gdz.top
Номер 635, страница 185 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
IV. Тригонометрия. 24. Основные тригонометрические тождества - номер 635, страница 185.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-635" "field_display_title" => "635" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:2 [ 0 => App\Models\Branch {#1069 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1102 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107666 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. Основные тригонометрические тождества" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "183" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107666 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. Основные тригонометрические тождества" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "183" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:3 [ 0 => App\Models\Element {#1150 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276990 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Edition {#1151 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108371" "type" => "task" ] "text" => "<p><strong>635.</strong> Упростите выражение:</p><p><strong>а)</strong> $\frac{\sin^2 \alpha}{1 - \cos \alpha}$</p><p><strong>б)</strong> $\frac{\cos^2 \alpha}{\sin \alpha + 1}$</p><p><strong>в)</strong> $\frac{1 - 2\sin^2 \alpha}{\cos \alpha + \sin \alpha}$</p><p><strong>г)</strong> $\frac{\sin \alpha - \cos \alpha}{1 - 2\cos^2 \alpha}$</p><p><strong>д)</strong> $\frac{\cos \alpha}{1 - \sin \alpha} - \frac{\cos \alpha}{1 + \sin \alpha}$</p><p><strong>е)</strong> $\frac{\sin^2 \alpha - \operatorname{tg}^2 \alpha}{\cos^2 \alpha - \operatorname{ctg}^2 \alpha}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "635-1.jpg" "alt" => null "width" => "1469" "height" => 395 "path" => "/media/algebra_09/soltan-u19/0-1/635-1.webp?ts=1752834644" ] ] ] #original: array:7 [ "id" => 1276990 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "1108371" "type" => "task" ] "text" => "<p><strong>635.</strong> Упростите выражение:</p><p><strong>а)</strong> $\frac{\sin^2 \alpha}{1 - \cos \alpha}$</p><p><strong>б)</strong> $\frac{\cos^2 \alpha}{\sin \alpha + 1}$</p><p><strong>в)</strong> $\frac{1 - 2\sin^2 \alpha}{\cos \alpha + \sin \alpha}$</p><p><strong>г)</strong> $\frac{\sin \alpha - \cos \alpha}{1 - 2\cos^2 \alpha}$</p><p><strong>д)</strong> $\frac{\cos \alpha}{1 - \sin \alpha} - \frac{\cos \alpha}{1 + \sin \alpha}$</p><p><strong>е)</strong> $\frac{\sin^2 \alpha - \operatorname{tg}^2 \alpha}{\cos^2 \alpha - \operatorname{ctg}^2 \alpha}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "635-1.jpg" "alt" => null "width" => "1469" "height" => 395 "path" => "/media/algebra_09/soltan-u19/0-1/635-1.webp?ts=1752834644" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1157 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278281 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Edition {#1158 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108371" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "635-1.jpg" "alt" => null "width" => "1436" "height" => 1958 "path" => "/media/algebra_09/soltan-u19/1-1/635-1.webp?ts=1752831420" ] ] ] #original: array:6 [ "id" => 1278281 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167} "task" => array:2 [ "refs" => "1108371" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "635-1.jpg" "alt" => null "width" => "1436" "height" => 1958 "path" => "/media/algebra_09/soltan-u19/1-1/635-1.webp?ts=1752831420" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1165 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554236 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1166 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108371" "type" => "task" ] "text" => "<p><strong>а)</strong> Для упрощения выражения $\frac{\sin^2 \alpha}{1 - \cos \alpha}$ воспользуемся основным тригонометрическим тождеством $\sin^2 \alpha + \cos^2 \alpha = 1$, из которого $\sin^2 \alpha = 1 - \cos^2 \alpha$.<br>Подставим это в числитель: $\frac{1 - \cos^2 \alpha}{1 - \cos \alpha}$<br>Применим формулу разности квадратов $a^2 - b^2 = (a-b)(a+b)$ к числителю: $\frac{(1 - \cos \alpha)(1 + \cos \alpha)}{1 - \cos \alpha}$<br>Сократим дробь на общий множитель $(1 - \cos \alpha)$: $1 + \cos \alpha$<br>Ответ: $1 + \cos \alpha$</p><p><strong>б)</strong> Для упрощения выражения $\frac{\cos^2 \alpha}{\sin \alpha + 1}$ воспользуемся основным тригонометрическим тождеством $\sin^2 \alpha + \cos^2 \alpha = 1$, из которого $\cos^2 \alpha = 1 - \sin^2 \alpha$.<br>Подставим это в числитель: $\frac{1 - \sin^2 \alpha}{1 + \sin \alpha}$<br>Применим формулу разности квадратов к числителю: $\frac{(1 - \sin \alpha)(1 + \sin \alpha)}{1 + \sin \alpha}$<br>Сократим дробь на общий множитель $(1 + \sin \alpha)$: $1 - \sin \alpha$<br>Ответ: $1 - \sin \alpha$</p><p><strong>в)</strong> Для упрощения выражения $\frac{1 - 2\sin^2 \alpha}{\cos \alpha + \sin \alpha}$ преобразуем числитель. Используем основное тригонометрическое тождество, представив $1 = \sin^2 \alpha + \cos^2 \alpha$:<br>$1 - 2\sin^2 \alpha = (\sin^2 \alpha + \cos^2 \alpha) - 2\sin^2 \alpha = \cos^2 \alpha - \sin^2 \alpha$<br>Теперь выражение выглядит так: $\frac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha + \sin \alpha}$<br>Применим формулу разности квадратов к числителю: $\frac{(\cos \alpha - \sin \alpha)(\cos \alpha + \sin \alpha)}{\cos \alpha + \sin \alpha}$<br>Сократим дробь на общий множитель $(\cos \alpha + \sin \alpha)$: $\cos \alpha - \sin \alpha$<br>Ответ: $\cos \alpha - \sin \alpha$</p><p><strong>г)</strong> Для упрощения выражения $\frac{\sin \alpha - \cos \alpha}{1 - 2\cos^2 \alpha}$ преобразуем знаменатель, используя $1 = \sin^2 \alpha + \cos^2 \alpha$:<br>$1 - 2\cos^2 \alpha = (\sin^2 \alpha + \cos^2 \alpha) - 2\cos^2 \alpha = \sin^2 \alpha - \cos^2 \alpha$<br>Теперь выражение выглядит так: $\frac{\sin \alpha - \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$<br>Применим формулу разности квадратов к знаменателю: $\frac{\sin \alpha - \cos \alpha}{(\sin \alpha - \cos \alpha)(\sin \alpha + \cos \alpha)}$<br>Сократим дробь на общий множитель $(\sin \alpha - \cos \alpha)$: $\frac{1}{\sin \alpha + \cos \alpha}$<br>Ответ: $\frac{1}{\sin \alpha + \cos \alpha}$</p><p><strong>д)</strong> Для упрощения выражения $\frac{\cos \alpha}{1 - \sin \alpha} - \frac{\cos \alpha}{1 + \sin \alpha}$ приведем дроби к общему знаменателю $(1 - \sin \alpha)(1 + \sin \alpha)$:<br>$\frac{\cos \alpha(1 + \sin \alpha) - \cos \alpha(1 - \sin \alpha)}{(1 - \sin \alpha)(1 + \sin \alpha)}$<br>Раскроем скобки в числителе: $\frac{\cos \alpha + \cos \alpha \sin \alpha - \cos \alpha + \cos \alpha \sin \alpha}{1 - \sin^2 \alpha}$<br>Упростим числитель и применим основное тригонометрическое тождество к знаменателю ($1 - \sin^2 \alpha = \cos^2 \alpha$): $\frac{2\sin \alpha \cos \alpha}{\cos^2 \alpha}$<br>Сократим дробь на $\cos \alpha$: $\frac{2\sin \alpha}{\cos \alpha}$<br>Используя определение тангенса $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$, получаем: $2\tan \alpha$<br>Ответ: $2\tan \alpha$</p><p><strong>е)</strong> Для упрощения выражения $\frac{\sin^2 \alpha - \tan^2 \alpha}{\cos^2 \alpha - \cot^2 \alpha}$ заменим тангенс и котангенс через синус и косинус: $\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}$ и $\cot^2 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha}$.<br>$\frac{\sin^2 \alpha - \frac{\sin^2 \alpha}{\cos^2 \alpha}}{\cos^2 \alpha - \frac{\cos^2 \alpha}{\sin^2 \alpha}}$<br>Вынесем общие множители в числителе и знаменателе: $\frac{\sin^2 \alpha (1 - \frac{1}{\cos^2 \alpha})}{\cos^2 \alpha (1 - \frac{1}{\sin^2 \alpha})}$<br>Приведем к общему знаменателю выражения в скобках: $\frac{\sin^2 \alpha (\frac{\cos^2 \alpha - 1}{\cos^2 \alpha})}{\cos^2 \alpha (\frac{\sin^2 \alpha - 1}{\sin^2 \alpha})}$<br>Из основного тождества $\sin^2 \alpha + \cos^2 \alpha = 1$ следует, что $\cos^2 \alpha - 1 = -\sin^2 \alpha$ и $\sin^2 \alpha - 1 = -\cos^2 \alpha$. Подставим: $\frac{\sin^2 \alpha (\frac{-\sin^2 \alpha}{\cos^2 \alpha})}{\cos^2 \alpha (\frac{-\cos^2 \alpha}{\sin^2 \alpha})} = \frac{-\frac{\sin^4 \alpha}{\cos^2 \alpha}}{-\frac{\cos^4 \alpha}{\sin^2 \alpha}}$<br>Упростим "четырехэтажную" дробь, умножив числитель на перевернутый знаменатель: $\frac{\sin^4 \alpha}{\cos^2 \alpha} \cdot \frac{\sin^2 \alpha}{\cos^4 \alpha} = \frac{\sin^6 \alpha}{\cos^6 \alpha}$<br>Используя определение тангенса, получаем: $\tan^6 \alpha$<br>Ответ: $\tan^6 \alpha$</p>" ] #original: array:6 [ "id" => 1554236 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1108371" "type" => "task" ] "text" => "<p><strong>а)</strong> Для упрощения выражения $\frac{\sin^2 \alpha}{1 - \cos \alpha}$ воспользуемся основным тригонометрическим тождеством $\sin^2 \alpha + \cos^2 \alpha = 1$, из которого $\sin^2 \alpha = 1 - \cos^2 \alpha$.<br>Подставим это в числитель: $\frac{1 - \cos^2 \alpha}{1 - \cos \alpha}$<br>Применим формулу разности квадратов $a^2 - b^2 = (a-b)(a+b)$ к числителю: $\frac{(1 - \cos \alpha)(1 + \cos \alpha)}{1 - \cos \alpha}$<br>Сократим дробь на общий множитель $(1 - \cos \alpha)$: $1 + \cos \alpha$<br>Ответ: $1 + \cos \alpha$</p><p><strong>б)</strong> Для упрощения выражения $\frac{\cos^2 \alpha}{\sin \alpha + 1}$ воспользуемся основным тригонометрическим тождеством $\sin^2 \alpha + \cos^2 \alpha = 1$, из которого $\cos^2 \alpha = 1 - \sin^2 \alpha$.<br>Подставим это в числитель: $\frac{1 - \sin^2 \alpha}{1 + \sin \alpha}$<br>Применим формулу разности квадратов к числителю: $\frac{(1 - \sin \alpha)(1 + \sin \alpha)}{1 + \sin \alpha}$<br>Сократим дробь на общий множитель $(1 + \sin \alpha)$: $1 - \sin \alpha$<br>Ответ: $1 - \sin \alpha$</p><p><strong>в)</strong> Для упрощения выражения $\frac{1 - 2\sin^2 \alpha}{\cos \alpha + \sin \alpha}$ преобразуем числитель. Используем основное тригонометрическое тождество, представив $1 = \sin^2 \alpha + \cos^2 \alpha$:<br>$1 - 2\sin^2 \alpha = (\sin^2 \alpha + \cos^2 \alpha) - 2\sin^2 \alpha = \cos^2 \alpha - \sin^2 \alpha$<br>Теперь выражение выглядит так: $\frac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha + \sin \alpha}$<br>Применим формулу разности квадратов к числителю: $\frac{(\cos \alpha - \sin \alpha)(\cos \alpha + \sin \alpha)}{\cos \alpha + \sin \alpha}$<br>Сократим дробь на общий множитель $(\cos \alpha + \sin \alpha)$: $\cos \alpha - \sin \alpha$<br>Ответ: $\cos \alpha - \sin \alpha$</p><p><strong>г)</strong> Для упрощения выражения $\frac{\sin \alpha - \cos \alpha}{1 - 2\cos^2 \alpha}$ преобразуем знаменатель, используя $1 = \sin^2 \alpha + \cos^2 \alpha$:<br>$1 - 2\cos^2 \alpha = (\sin^2 \alpha + \cos^2 \alpha) - 2\cos^2 \alpha = \sin^2 \alpha - \cos^2 \alpha$<br>Теперь выражение выглядит так: $\frac{\sin \alpha - \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$<br>Применим формулу разности квадратов к знаменателю: $\frac{\sin \alpha - \cos \alpha}{(\sin \alpha - \cos \alpha)(\sin \alpha + \cos \alpha)}$<br>Сократим дробь на общий множитель $(\sin \alpha - \cos \alpha)$: $\frac{1}{\sin \alpha + \cos \alpha}$<br>Ответ: $\frac{1}{\sin \alpha + \cos \alpha}$</p><p><strong>д)</strong> Для упрощения выражения $\frac{\cos \alpha}{1 - \sin \alpha} - \frac{\cos \alpha}{1 + \sin \alpha}$ приведем дроби к общему знаменателю $(1 - \sin \alpha)(1 + \sin \alpha)$:<br>$\frac{\cos \alpha(1 + \sin \alpha) - \cos \alpha(1 - \sin \alpha)}{(1 - \sin \alpha)(1 + \sin \alpha)}$<br>Раскроем скобки в числителе: $\frac{\cos \alpha + \cos \alpha \sin \alpha - \cos \alpha + \cos \alpha \sin \alpha}{1 - \sin^2 \alpha}$<br>Упростим числитель и применим основное тригонометрическое тождество к знаменателю ($1 - \sin^2 \alpha = \cos^2 \alpha$): $\frac{2\sin \alpha \cos \alpha}{\cos^2 \alpha}$<br>Сократим дробь на $\cos \alpha$: $\frac{2\sin \alpha}{\cos \alpha}$<br>Используя определение тангенса $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$, получаем: $2\tan \alpha$<br>Ответ: $2\tan \alpha$</p><p><strong>е)</strong> Для упрощения выражения $\frac{\sin^2 \alpha - \tan^2 \alpha}{\cos^2 \alpha - \cot^2 \alpha}$ заменим тангенс и котангенс через синус и косинус: $\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}$ и $\cot^2 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha}$.<br>$\frac{\sin^2 \alpha - \frac{\sin^2 \alpha}{\cos^2 \alpha}}{\cos^2 \alpha - \frac{\cos^2 \alpha}{\sin^2 \alpha}}$<br>Вынесем общие множители в числителе и знаменателе: $\frac{\sin^2 \alpha (1 - \frac{1}{\cos^2 \alpha})}{\cos^2 \alpha (1 - \frac{1}{\sin^2 \alpha})}$<br>Приведем к общему знаменателю выражения в скобках: $\frac{\sin^2 \alpha (\frac{\cos^2 \alpha - 1}{\cos^2 \alpha})}{\cos^2 \alpha (\frac{\sin^2 \alpha - 1}{\sin^2 \alpha})}$<br>Из основного тождества $\sin^2 \alpha + \cos^2 \alpha = 1$ следует, что $\cos^2 \alpha - 1 = -\sin^2 \alpha$ и $\sin^2 \alpha - 1 = -\cos^2 \alpha$. Подставим: $\frac{\sin^2 \alpha (\frac{-\sin^2 \alpha}{\cos^2 \alpha})}{\cos^2 \alpha (\frac{-\cos^2 \alpha}{\sin^2 \alpha})} = \frac{-\frac{\sin^4 \alpha}{\cos^2 \alpha}}{-\frac{\cos^4 \alpha}{\sin^2 \alpha}}$<br>Упростим "четырехэтажную" дробь, умножив числитель на перевернутый знаменатель: $\frac{\sin^4 \alpha}{\cos^2 \alpha} \cdot \frac{\sin^2 \alpha}{\cos^4 \alpha} = \frac{\sin^6 \alpha}{\cos^6 \alpha}$<br>Используя определение тангенса, получаем: $\tan^6 \alpha$<br>Ответ: $\tan^6 \alpha$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108372" "type" => "task" ] "previous" => array:2 [ "refs" => "1108370" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1147 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Term {#1139 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1148 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Term {#1142 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:3 [ 0 => App\Models\Term {#1143 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1173 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1174 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Term {#1178 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Term {#1180 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Term {#1182 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1140} "field_class" => Illuminate\Database\Eloquent\Collection {#1146} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1144} "field_author" => Illuminate\Database\Eloquent\Collection {#1145} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1176} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_country" => Illuminate\Database\Eloquent\Collection {#1179} "field_city" => Illuminate\Database\Eloquent\Collection {#1181} "field_series" => Illuminate\Database\Eloquent\Collection {#1183} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\BookPage {#1199 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097927 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-185" "field_display_title" => "185" "field_folder" => "folder1" "field_image_name" => "185" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Book {#1147} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1203 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Element {#1212 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261078 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261078 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097928" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097926" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1370 #items: array:6 [ 0 => App\Models\Task {#1382 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108368 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-632" "field_display_title" => "632" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1384 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1385 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1386 …2} "content" => Illuminate\Database\Eloquent\Collection {#1412 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1413 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108368 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-632" "field_display_title" => "632" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1384 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1385 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1386 …2} "content" => Illuminate\Database\Eloquent\Collection {#1412 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1413 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1411 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108369 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-633" "field_display_title" => "633" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1414 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1415 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1416 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1417 …2} "content" => Illuminate\Database\Eloquent\Collection {#1420 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108369 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-633" "field_display_title" => "633" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1414 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1415 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1416 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1417 …2} "content" => Illuminate\Database\Eloquent\Collection {#1420 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1419 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-634" "field_display_title" => "634" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1418 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1425 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1427 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1423 …2} "content" => Illuminate\Database\Eloquent\Collection {#1422 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1440 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-634" "field_display_title" => "634" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1418 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1425 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1427 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1423 …2} "content" => Illuminate\Database\Eloquent\Collection {#1422 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1440 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1424 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-635" "field_display_title" => "635" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1421 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1439 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1441 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1437 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1438 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-635" "field_display_title" => "635" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1421 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1439 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1441 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1437 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1438 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1436 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-636" "field_display_title" => "636" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1449 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1450 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1451 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1452 …2} "content" => Illuminate\Database\Eloquent\Collection {#1455 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1461 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-636" "field_display_title" => "636" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1449 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1450 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1451 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1452 …2} "content" => Illuminate\Database\Eloquent\Collection {#1455 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1461 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1454 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-637" "field_display_title" => "637" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1453 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1460 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1462 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1458 …2} "content" => Illuminate\Database\Eloquent\Collection {#1457 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1475 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-637" "field_display_title" => "637" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1453 …2} "field_metatags_title" => null "field_metatags_description" => null …13 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097927 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-185" "field_display_title" => "185" "field_folder" => "folder1" "field_image_name" => "185" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1203} "content" => Illuminate\Database\Eloquent\Collection {#1213} "next" => array:2 [ "refs" => "1097928" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097926" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1370} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1108371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-635" "field_display_title" => "635" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1138} "content" => Illuminate\Database\Eloquent\Collection {#1141} "next" => array:2 [ "refs" => "1108372" "type" => "task" ] "previous" => array:2 [ "refs" => "1108370" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195} "page" => array:2 [ "refs" => "1097927" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№635 (с. 185)
Условие. №635 (с. 185)
скриншот условия
635. Упростите выражение:
а) $\frac{\sin^2 \alpha}{1 - \cos \alpha}$
б) $\frac{\cos^2 \alpha}{\sin \alpha + 1}$
в) $\frac{1 - 2\sin^2 \alpha}{\cos \alpha + \sin \alpha}$
г) $\frac{\sin \alpha - \cos \alpha}{1 - 2\cos^2 \alpha}$
д) $\frac{\cos \alpha}{1 - \sin \alpha} - \frac{\cos \alpha}{1 + \sin \alpha}$
е) $\frac{\sin^2 \alpha - \operatorname{tg}^2 \alpha}{\cos^2 \alpha - \operatorname{ctg}^2 \alpha}$
Решение. №635 (с. 185)
Решение 2 (rus). №635 (с. 185)
а) Для упрощения выражения $\frac{\sin^2 \alpha}{1 - \cos \alpha}$ воспользуемся основным тригонометрическим тождеством $\sin^2 \alpha + \cos^2 \alpha = 1$, из которого $\sin^2 \alpha = 1 - \cos^2 \alpha$.
Подставим это в числитель: $\frac{1 - \cos^2 \alpha}{1 - \cos \alpha}$
Применим формулу разности квадратов $a^2 - b^2 = (a-b)(a+b)$ к числителю: $\frac{(1 - \cos \alpha)(1 + \cos \alpha)}{1 - \cos \alpha}$
Сократим дробь на общий множитель $(1 - \cos \alpha)$: $1 + \cos \alpha$
Ответ: $1 + \cos \alpha$
б) Для упрощения выражения $\frac{\cos^2 \alpha}{\sin \alpha + 1}$ воспользуемся основным тригонометрическим тождеством $\sin^2 \alpha + \cos^2 \alpha = 1$, из которого $\cos^2 \alpha = 1 - \sin^2 \alpha$.
Подставим это в числитель: $\frac{1 - \sin^2 \alpha}{1 + \sin \alpha}$
Применим формулу разности квадратов к числителю: $\frac{(1 - \sin \alpha)(1 + \sin \alpha)}{1 + \sin \alpha}$
Сократим дробь на общий множитель $(1 + \sin \alpha)$: $1 - \sin \alpha$
Ответ: $1 - \sin \alpha$
в) Для упрощения выражения $\frac{1 - 2\sin^2 \alpha}{\cos \alpha + \sin \alpha}$ преобразуем числитель. Используем основное тригонометрическое тождество, представив $1 = \sin^2 \alpha + \cos^2 \alpha$:
$1 - 2\sin^2 \alpha = (\sin^2 \alpha + \cos^2 \alpha) - 2\sin^2 \alpha = \cos^2 \alpha - \sin^2 \alpha$
Теперь выражение выглядит так: $\frac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha + \sin \alpha}$
Применим формулу разности квадратов к числителю: $\frac{(\cos \alpha - \sin \alpha)(\cos \alpha + \sin \alpha)}{\cos \alpha + \sin \alpha}$
Сократим дробь на общий множитель $(\cos \alpha + \sin \alpha)$: $\cos \alpha - \sin \alpha$
Ответ: $\cos \alpha - \sin \alpha$
г) Для упрощения выражения $\frac{\sin \alpha - \cos \alpha}{1 - 2\cos^2 \alpha}$ преобразуем знаменатель, используя $1 = \sin^2 \alpha + \cos^2 \alpha$:
$1 - 2\cos^2 \alpha = (\sin^2 \alpha + \cos^2 \alpha) - 2\cos^2 \alpha = \sin^2 \alpha - \cos^2 \alpha$
Теперь выражение выглядит так: $\frac{\sin \alpha - \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$
Применим формулу разности квадратов к знаменателю: $\frac{\sin \alpha - \cos \alpha}{(\sin \alpha - \cos \alpha)(\sin \alpha + \cos \alpha)}$
Сократим дробь на общий множитель $(\sin \alpha - \cos \alpha)$: $\frac{1}{\sin \alpha + \cos \alpha}$
Ответ: $\frac{1}{\sin \alpha + \cos \alpha}$
д) Для упрощения выражения $\frac{\cos \alpha}{1 - \sin \alpha} - \frac{\cos \alpha}{1 + \sin \alpha}$ приведем дроби к общему знаменателю $(1 - \sin \alpha)(1 + \sin \alpha)$:
$\frac{\cos \alpha(1 + \sin \alpha) - \cos \alpha(1 - \sin \alpha)}{(1 - \sin \alpha)(1 + \sin \alpha)}$
Раскроем скобки в числителе: $\frac{\cos \alpha + \cos \alpha \sin \alpha - \cos \alpha + \cos \alpha \sin \alpha}{1 - \sin^2 \alpha}$
Упростим числитель и применим основное тригонометрическое тождество к знаменателю ($1 - \sin^2 \alpha = \cos^2 \alpha$): $\frac{2\sin \alpha \cos \alpha}{\cos^2 \alpha}$
Сократим дробь на $\cos \alpha$: $\frac{2\sin \alpha}{\cos \alpha}$
Используя определение тангенса $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$, получаем: $2\tan \alpha$
Ответ: $2\tan \alpha$
е) Для упрощения выражения $\frac{\sin^2 \alpha - \tan^2 \alpha}{\cos^2 \alpha - \cot^2 \alpha}$ заменим тангенс и котангенс через синус и косинус: $\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}$ и $\cot^2 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha}$.
$\frac{\sin^2 \alpha - \frac{\sin^2 \alpha}{\cos^2 \alpha}}{\cos^2 \alpha - \frac{\cos^2 \alpha}{\sin^2 \alpha}}$
Вынесем общие множители в числителе и знаменателе: $\frac{\sin^2 \alpha (1 - \frac{1}{\cos^2 \alpha})}{\cos^2 \alpha (1 - \frac{1}{\sin^2 \alpha})}$
Приведем к общему знаменателю выражения в скобках: $\frac{\sin^2 \alpha (\frac{\cos^2 \alpha - 1}{\cos^2 \alpha})}{\cos^2 \alpha (\frac{\sin^2 \alpha - 1}{\sin^2 \alpha})}$
Из основного тождества $\sin^2 \alpha + \cos^2 \alpha = 1$ следует, что $\cos^2 \alpha - 1 = -\sin^2 \alpha$ и $\sin^2 \alpha - 1 = -\cos^2 \alpha$. Подставим: $\frac{\sin^2 \alpha (\frac{-\sin^2 \alpha}{\cos^2 \alpha})}{\cos^2 \alpha (\frac{-\cos^2 \alpha}{\sin^2 \alpha})} = \frac{-\frac{\sin^4 \alpha}{\cos^2 \alpha}}{-\frac{\cos^4 \alpha}{\sin^2 \alpha}}$
Упростим "четырехэтажную" дробь, умножив числитель на перевернутый знаменатель: $\frac{\sin^4 \alpha}{\cos^2 \alpha} \cdot \frac{\sin^2 \alpha}{\cos^4 \alpha} = \frac{\sin^6 \alpha}{\cos^6 \alpha}$
Используя определение тангенса, получаем: $\tan^6 \alpha$
Ответ: $\tan^6 \alpha$
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по алгебре за 9 класс, для упражнения номер 635 расположенного на странице 185 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по алгебре к упражнению №635 (с. 185), авторов: Солтан (Генадий Николаевич), Солтан (Алла Евгеньевна), Жумадилова (Аманбала Жумадиловна), учебного пособия издательства Кокшетау.