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Номер 770, страница 218 - 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. 29. Преобразование произведения тригонометрических функций в сумму и разность - номер 770, страница 218.
App\Models\Task {#1024 // 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" => 1108513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-770" "field_display_title" => "770" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1060 #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 {#333} "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #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 {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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" => 1107671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "29. Преобразование произведения тригонометрических функций в сумму и разность" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "216" "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 {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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{#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …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 {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #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 {#1095 …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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1095 …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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …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" => 1107671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "29. Преобразование произведения тригонометрических функций в сумму и разность" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "216" "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #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" => 1277125 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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" => "1108513" "type" => "task" ] "text" => "<p><strong>770.</strong> Докажите, что верно равенство:</p><p><strong>a)</strong> $\operatorname{tg} 20^\circ + \operatorname{tg} 25^\circ + \operatorname{tg} 20^\circ \cdot \operatorname{tg} 25^\circ = 1;$</p><p><strong>б)</strong> $\operatorname{ctg} 10^\circ - \operatorname{tg} 35^\circ - \operatorname{ctg} 10^\circ \cdot \operatorname{tg} 35^\circ = 1;$</p><p><strong>в)</strong> $\cos 15^\circ \cdot \cos 7^\circ - \sin 79^\circ \cdot \cos 11^\circ - \sin 86^\circ \cdot \cos 4^\circ = -1;$</p><p><strong>г)</strong> $\sin 17^\circ \cdot \sin 73^\circ - \sin 21^\circ \cdot \cos 13^\circ + \sin 4^\circ \cdot \sin 86^\circ = 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "770-1.jpg" "alt" => null "width" => "1413" "height" => 384 "path" => "/media/algebra_09/soltan-u19/0-1/770-1.webp?ts=1752834761" ] ] ] #original: array:7 [ "id" => 1277125 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108513" "type" => "task" ] "text" => "<p><strong>770.</strong> Докажите, что верно равенство:</p><p><strong>a)</strong> $\operatorname{tg} 20^\circ + \operatorname{tg} 25^\circ + \operatorname{tg} 20^\circ \cdot \operatorname{tg} 25^\circ = 1;$</p><p><strong>б)</strong> $\operatorname{ctg} 10^\circ - \operatorname{tg} 35^\circ - \operatorname{ctg} 10^\circ \cdot \operatorname{tg} 35^\circ = 1;$</p><p><strong>в)</strong> $\cos 15^\circ \cdot \cos 7^\circ - \sin 79^\circ \cdot \cos 11^\circ - \sin 86^\circ \cdot \cos 4^\circ = -1;$</p><p><strong>г)</strong> $\sin 17^\circ \cdot \sin 73^\circ - \sin 21^\circ \cdot \cos 13^\circ + \sin 4^\circ \cdot \sin 86^\circ = 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "770-1.jpg" "alt" => null "width" => "1413" "height" => 384 "path" => "/media/algebra_09/soltan-u19/0-1/770-1.webp?ts=1752834761" ] ] ] #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 {#1107 #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" => 1278416 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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" => "1108513" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "770-1.jpg" "alt" => null "width" => "1276" "height" => 1042 "path" => "/media/algebra_09/soltan-u19/1-1/770-1.webp?ts=1752831566" ] 1 => array:5 [ "name" => "770-2.jpg" "alt" => null "width" => "1403" "height" => 2387 "path" => "/media/algebra_09/soltan-u19/1-1/770-2.webp?ts=1752831566" ] ] ] #original: array:6 [ "id" => 1278416 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108513" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "770-1.jpg" "alt" => null "width" => "1276" "height" => 1042 "path" => "/media/algebra_09/soltan-u19/1-1/770-1.webp?ts=1752831566" ] 1 => array:5 [ "name" => "770-2.jpg" "alt" => null "width" => "1403" "height" => 2387 "path" => "/media/algebra_09/soltan-u19/1-1/770-2.webp?ts=1752831566" ] ] ] #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 {#1115 #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" => 1554371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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 {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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" => "1108513" "type" => "task" ] "text" => "<p><strong>а)</strong> Для доказательства воспользуемся формулой тангенса суммы: $\operatorname{tg}(\alpha + \beta) = \frac{\operatorname{tg}\alpha + \operatorname{tg}\beta}{1 - \operatorname{tg}\alpha \cdot \operatorname{tg}\beta}$.<br>Пусть $\alpha = 20^{\circ}$ и $\beta = 25^{\circ}$. Тогда их сумма $\alpha + \beta = 20^{\circ} + 25^{\circ} = 45^{\circ}$.<br>Мы знаем, что $\operatorname{tg}45^{\circ} = 1$.<br>Подставим значения в формулу:<br>$\operatorname{tg}(20^{\circ} + 25^{\circ}) = \frac{\operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}}{1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}}$<br>$1 = \frac{\operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}}{1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}}$<br>Умножим обе части на знаменатель $(1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ})$:<br>$1 \cdot (1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}) = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}$<br>$1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ} = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}$<br>Перенесем слагаемое $(-\operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ})$ в правую часть:<br>$1 = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ} + \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}$<br>Равенство доказано.<br>Ответ:</p><p><strong>б)</strong> Преобразуем левую часть равенства. Воспользуемся формулой приведения $\operatorname{ctg}\alpha = \operatorname{tg}(90^{\circ} - \alpha)$.<br>$\operatorname{ctg}10^{\circ} = \operatorname{tg}(90^{\circ} - 10^{\circ}) = \operatorname{tg}80^{\circ}$.<br>Теперь исходное равенство принимает вид:<br>$\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ} - \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ} = 1$.<br>Рассмотрим формулу тангенса разности: $\operatorname{tg}(\alpha - \beta) = \frac{\operatorname{tg}\alpha - \operatorname{tg}\beta}{1 + \operatorname{tg}\alpha \cdot \operatorname{tg}\beta}$.<br>Пусть $\alpha = 80^{\circ}$ и $\beta = 35^{\circ}$. Тогда их разность $\alpha - \beta = 80^{\circ} - 35^{\circ} = 45^{\circ}$.<br>Мы знаем, что $\operatorname{tg}45^{\circ} = 1$.<br>$\operatorname{tg}(80^{\circ} - 35^{\circ}) = \frac{\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}}{1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}}$<br>$1 = \frac{\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}}{1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}}$<br>Умножим обе части на знаменатель $(1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ})$:<br>$1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ} = \operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}$<br>Перенесем слагаемое $(\operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ})$ в правую часть:<br>$1 = \operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ} - \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}$<br>Подставив обратно $\operatorname{tg}80^{\circ} = \operatorname{ctg}10^{\circ}$, получаем исходное равенство. Равенство доказано.<br>Ответ:</p><p><strong>в)</strong> Преобразуем левую часть равенства, используя формулы приведения $\sin(90^{\circ} - \alpha) = \cos\alpha$:<br>$\sin 79^{\circ} = \sin(90^{\circ} - 11^{\circ}) = \cos 11^{\circ}$<br>$\sin 86^{\circ} = \sin(90^{\circ} - 4^{\circ}) = \cos 4^{\circ}$<br>Подставим эти выражения в исходное равенство:<br>$\cos 15^{\circ} \cdot \cos 7^{\circ} - \cos 11^{\circ} \cdot \cos 11^{\circ} - \cos 4^{\circ} \cdot \cos 4^{\circ} = \cos 15^{\circ} \cdot \cos 7^{\circ} - \cos^2 11^{\circ} - \cos^2 4^{\circ}$.<br>Теперь используем формулу произведения косинусов $\cos\alpha\cos\beta = \frac{1}{2}(\cos(\alpha+\beta) + \cos(\alpha-\beta))$ и формулу понижения степени $\cos^2\alpha = \frac{1+\cos(2\alpha)}{2}$.<br>$\cos 15^{\circ} \cdot \cos 7^{\circ} = \frac{1}{2}(\cos(15^{\circ}+7^{\circ}) + \cos(15^{\circ}-7^{\circ})) = \frac{1}{2}(\cos 22^{\circ} + \cos 8^{\circ})$<br>$\cos^2 11^{\circ} = \frac{1 + \cos(2 \cdot 11^{\circ})}{2} = \frac{1 + \cos 22^{\circ}}{2}$<br>$\cos^2 4^{\circ} = \frac{1 + \cos(2 \cdot 4^{\circ})}{2} = \frac{1 + \cos 8^{\circ}}{2}$<br>Подставим полученные выражения в левую часть:<br>$\frac{1}{2}(\cos 22^{\circ} + \cos 8^{\circ}) - \frac{1 + \cos 22^{\circ}}{2} - \frac{1 + \cos 8^{\circ}}{2}$<br>$= \frac{\cos 22^{\circ} + \cos 8^{\circ} - (1 + \cos 22^{\circ}) - (1 + \cos 8^{\circ})}{2}$<br>$= \frac{\cos 22^{\circ} + \cos 8^{\circ} - 1 - \cos 22^{\circ} - 1 - \cos 8^{\circ}}{2}$<br>$= \frac{(\cos 22^{\circ} - \cos 22^{\circ}) + (\cos 8^{\circ} - \cos 8^{\circ}) - 1 - 1}{2} = \frac{-2}{2} = -1$<br>Левая часть равна -1, что совпадает с правой частью. Равенство доказано.<br>Ответ:</p><p><strong>г)</strong> Преобразуем левую часть равенства, используя формулы приведения $\sin(90^{\circ} - \alpha) = \cos\alpha$:<br>$\sin 73^{\circ} = \sin(90^{\circ} - 17^{\circ}) = \cos 17^{\circ}$<br>$\sin 86^{\circ} = \sin(90^{\circ} - 4^{\circ}) = \cos 4^{\circ}$<br>Подставим эти выражения в исходное равенство:<br>$\sin 17^{\circ} \cdot \cos 17^{\circ} - \sin 21^{\circ} \cdot \cos 13^{\circ} + \sin 4^{\circ} \cdot \cos 4^{\circ}$.<br>Используем формулу синуса двойного угла $\sin(2\alpha) = 2\sin\alpha\cos\alpha$, откуда $\sin\alpha\cos\alpha = \frac{1}{2}\sin(2\alpha)$:<br>$\sin 17^{\circ} \cdot \cos 17^{\circ} = \frac{1}{2}\sin(2 \cdot 17^{\circ}) = \frac{1}{2}\sin 34^{\circ}$<br>$\sin 4^{\circ} \cdot \cos 4^{\circ} = \frac{1}{2}\sin(2 \cdot 4^{\circ}) = \frac{1}{2}\sin 8^{\circ}$<br>Подставим в выражение:<br>$\frac{1}{2}\sin 34^{\circ} - \sin 21^{\circ} \cdot \cos 13^{\circ} + \frac{1}{2}\sin 8^{\circ} = \frac{1}{2}(\sin 34^{\circ} + \sin 8^{\circ}) - \sin 21^{\circ} \cdot \cos 13^{\circ}$.<br>Теперь применим формулу суммы синусов $\sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$:<br>$\sin 34^{\circ} + \sin 8^{\circ} = 2\sin\frac{34^{\circ}+8^{\circ}}{2}\cos\frac{34^{\circ}-8^{\circ}}{2} = 2\sin\frac{42^{\circ}}{2}\cos\frac{26^{\circ}}{2} = 2\sin 21^{\circ}\cos 13^{\circ}$.<br>Подставим результат обратно в выражение:<br>$\frac{1}{2}(2\sin 21^{\circ}\cos 13^{\circ}) - \sin 21^{\circ} \cdot \cos 13^{\circ} = \sin 21^{\circ}\cos 13^{\circ} - \sin 21^{\circ}\cos 13^{\circ} = 0$.<br>Левая часть равна 0, что совпадает с правой частью. Равенство доказано.<br>Ответ:</p>" ] #original: array:6 [ "id" => 1554371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108513" "type" => "task" ] "text" => "<p><strong>а)</strong> Для доказательства воспользуемся формулой тангенса суммы: $\operatorname{tg}(\alpha + \beta) = \frac{\operatorname{tg}\alpha + \operatorname{tg}\beta}{1 - \operatorname{tg}\alpha \cdot \operatorname{tg}\beta}$.<br>Пусть $\alpha = 20^{\circ}$ и $\beta = 25^{\circ}$. Тогда их сумма $\alpha + \beta = 20^{\circ} + 25^{\circ} = 45^{\circ}$.<br>Мы знаем, что $\operatorname{tg}45^{\circ} = 1$.<br>Подставим значения в формулу:<br>$\operatorname{tg}(20^{\circ} + 25^{\circ}) = \frac{\operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}}{1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}}$<br>$1 = \frac{\operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}}{1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}}$<br>Умножим обе части на знаменатель $(1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ})$:<br>$1 \cdot (1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}) = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}$<br>$1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ} = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}$<br>Перенесем слагаемое $(-\operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ})$ в правую часть:<br>$1 = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ} + \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}$<br>Равенство доказано.<br>Ответ:</p><p><strong>б)</strong> Преобразуем левую часть равенства. Воспользуемся формулой приведения $\operatorname{ctg}\alpha = \operatorname{tg}(90^{\circ} - \alpha)$.<br>$\operatorname{ctg}10^{\circ} = \operatorname{tg}(90^{\circ} - 10^{\circ}) = \operatorname{tg}80^{\circ}$.<br>Теперь исходное равенство принимает вид:<br>$\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ} - \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ} = 1$.<br>Рассмотрим формулу тангенса разности: $\operatorname{tg}(\alpha - \beta) = \frac{\operatorname{tg}\alpha - \operatorname{tg}\beta}{1 + \operatorname{tg}\alpha \cdot \operatorname{tg}\beta}$.<br>Пусть $\alpha = 80^{\circ}$ и $\beta = 35^{\circ}$. Тогда их разность $\alpha - \beta = 80^{\circ} - 35^{\circ} = 45^{\circ}$.<br>Мы знаем, что $\operatorname{tg}45^{\circ} = 1$.<br>$\operatorname{tg}(80^{\circ} - 35^{\circ}) = \frac{\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}}{1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}}$<br>$1 = \frac{\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}}{1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}}$<br>Умножим обе части на знаменатель $(1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ})$:<br>$1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ} = \operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}$<br>Перенесем слагаемое $(\operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ})$ в правую часть:<br>$1 = \operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ} - \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}$<br>Подставив обратно $\operatorname{tg}80^{\circ} = \operatorname{ctg}10^{\circ}$, получаем исходное равенство. Равенство доказано.<br>Ответ:</p><p><strong>в)</strong> Преобразуем левую часть равенства, используя формулы приведения $\sin(90^{\circ} - \alpha) = \cos\alpha$:<br>$\sin 79^{\circ} = \sin(90^{\circ} - 11^{\circ}) = \cos 11^{\circ}$<br>$\sin 86^{\circ} = \sin(90^{\circ} - 4^{\circ}) = \cos 4^{\circ}$<br>Подставим эти выражения в исходное равенство:<br>$\cos 15^{\circ} \cdot \cos 7^{\circ} - \cos 11^{\circ} \cdot \cos 11^{\circ} - \cos 4^{\circ} \cdot \cos 4^{\circ} = \cos 15^{\circ} \cdot \cos 7^{\circ} - \cos^2 11^{\circ} - \cos^2 4^{\circ}$.<br>Теперь используем формулу произведения косинусов $\cos\alpha\cos\beta = \frac{1}{2}(\cos(\alpha+\beta) + \cos(\alpha-\beta))$ и формулу понижения степени $\cos^2\alpha = \frac{1+\cos(2\alpha)}{2}$.<br>$\cos 15^{\circ} \cdot \cos 7^{\circ} = \frac{1}{2}(\cos(15^{\circ}+7^{\circ}) + \cos(15^{\circ}-7^{\circ})) = \frac{1}{2}(\cos 22^{\circ} + \cos 8^{\circ})$<br>$\cos^2 11^{\circ} = \frac{1 + \cos(2 \cdot 11^{\circ})}{2} = \frac{1 + \cos 22^{\circ}}{2}$<br>$\cos^2 4^{\circ} = \frac{1 + \cos(2 \cdot 4^{\circ})}{2} = \frac{1 + \cos 8^{\circ}}{2}$<br>Подставим полученные выражения в левую часть:<br>$\frac{1}{2}(\cos 22^{\circ} + \cos 8^{\circ}) - \frac{1 + \cos 22^{\circ}}{2} - \frac{1 + \cos 8^{\circ}}{2}$<br>$= \frac{\cos 22^{\circ} + \cos 8^{\circ} - (1 + \cos 22^{\circ}) - (1 + \cos 8^{\circ})}{2}$<br>$= \frac{\cos 22^{\circ} + \cos 8^{\circ} - 1 - \cos 22^{\circ} - 1 - \cos 8^{\circ}}{2}$<br>$= \frac{(\cos 22^{\circ} - \cos 22^{\circ}) + (\cos 8^{\circ} - \cos 8^{\circ}) - 1 - 1}{2} = \frac{-2}{2} = -1$<br>Левая часть равна -1, что совпадает с правой частью. Равенство доказано.<br>Ответ:</p><p><strong>г)</strong> Преобразуем левую часть равенства, используя формулы приведения $\sin(90^{\circ} - \alpha) = \cos\alpha$:<br>$\sin 73^{\circ} = \sin(90^{\circ} - 17^{\circ}) = \cos 17^{\circ}$<br>$\sin 86^{\circ} = \sin(90^{\circ} - 4^{\circ}) = \cos 4^{\circ}$<br>Подставим эти выражения в исходное равенство:<br>$\sin 17^{\circ} \cdot \cos 17^{\circ} - \sin 21^{\circ} \cdot \cos 13^{\circ} + \sin 4^{\circ} \cdot \cos 4^{\circ}$.<br>Используем формулу синуса двойного угла $\sin(2\alpha) = 2\sin\alpha\cos\alpha$, откуда $\sin\alpha\cos\alpha = \frac{1}{2}\sin(2\alpha)$:<br>$\sin 17^{\circ} \cdot \cos 17^{\circ} = \frac{1}{2}\sin(2 \cdot 17^{\circ}) = \frac{1}{2}\sin 34^{\circ}$<br>$\sin 4^{\circ} \cdot \cos 4^{\circ} = \frac{1}{2}\sin(2 \cdot 4^{\circ}) = \frac{1}{2}\sin 8^{\circ}$<br>Подставим в выражение:<br>$\frac{1}{2}\sin 34^{\circ} - \sin 21^{\circ} \cdot \cos 13^{\circ} + \frac{1}{2}\sin 8^{\circ} = \frac{1}{2}(\sin 34^{\circ} + \sin 8^{\circ}) - \sin 21^{\circ} \cdot \cos 13^{\circ}$.<br>Теперь применим формулу суммы синусов $\sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$:<br>$\sin 34^{\circ} + \sin 8^{\circ} = 2\sin\frac{34^{\circ}+8^{\circ}}{2}\cos\frac{34^{\circ}-8^{\circ}}{2} = 2\sin\frac{42^{\circ}}{2}\cos\frac{26^{\circ}}{2} = 2\sin 21^{\circ}\cos 13^{\circ}$.<br>Подставим результат обратно в выражение:<br>$\frac{1}{2}(2\sin 21^{\circ}\cos 13^{\circ}) - \sin 21^{\circ} \cdot \cos 13^{\circ} = \sin 21^{\circ}\cos 13^{\circ} - \sin 21^{\circ}\cos 13^{\circ} = 0$.<br>Левая часть равна 0, что совпадает с правой частью. Равенство доказано.<br>Ответ:</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" => "1108514" "type" => "task" ] "previous" => array:2 [ "refs" => "1108512" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #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" => 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"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 {#1125} "field_class" => Illuminate\Database\Eloquent\Collection {#1127} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129} "field_author" => Illuminate\Database\Eloquent\Collection {#1131} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_country" => Illuminate\Database\Eloquent\Collection {#1138} "field_city" => Illuminate\Database\Eloquent\Collection {#1140} "field_series" => Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "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 {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "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 {#1166 #items: array:1 [ 0 => App\Models\BookPage {#1165 #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" => 1097960 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-218" "field_display_title" => "218" "field_folder" => "folder1" "field_image_name" => "218" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1168 #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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 } "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 {#1196 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:1 [ 0 => App\Models\Element {#1206 #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" => 1261111 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261111 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …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" => "1097961" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097959" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1969 #items: array:7 [ 0 => App\Models\Task {#1199 #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" => 1108511 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => "218" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-768" "field_display_title" => "768" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1203 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 …2} "content" => Illuminate\Database\Eloquent\Collection {#1254 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1279 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108511 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => "218" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-768" "field_display_title" => "768" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1203 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 …2} "content" => Illuminate\Database\Eloquent\Collection {#1254 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1279 …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 {#1280 #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" => 1108512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-769" "field_display_title" => "769" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1281 …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 {#1283 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1317 …2} "content" => Illuminate\Database\Eloquent\Collection {#1354 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1379 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-769" "field_display_title" => "769" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1281 …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 {#1283 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1317 …2} "content" => Illuminate\Database\Eloquent\Collection {#1354 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1379 …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 {#1408 #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" => 1108513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-770" "field_display_title" => "770" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1409 …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 {#1411 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1412 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1445 …2} "content" => Illuminate\Database\Eloquent\Collection {#1482 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1507 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-770" "field_display_title" => "770" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1409 …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 {#1411 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1412 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1445 …2} "content" => Illuminate\Database\Eloquent\Collection {#1482 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1507 …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 {#1536 #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" => 1108514 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-771" "field_display_title" => "771" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1537 …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 {#1539 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1540 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1573 …2} "content" => Illuminate\Database\Eloquent\Collection {#1610 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1635 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 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[] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1664 #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" => 1108515 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-772" "field_display_title" => "772" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1665 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null 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№770 (с. 218)
Условие. №770 (с. 218)
скриншот условия
770. Докажите, что верно равенство:
a) $\operatorname{tg} 20^\circ + \operatorname{tg} 25^\circ + \operatorname{tg} 20^\circ \cdot \operatorname{tg} 25^\circ = 1;$
б) $\operatorname{ctg} 10^\circ - \operatorname{tg} 35^\circ - \operatorname{ctg} 10^\circ \cdot \operatorname{tg} 35^\circ = 1;$
в) $\cos 15^\circ \cdot \cos 7^\circ - \sin 79^\circ \cdot \cos 11^\circ - \sin 86^\circ \cdot \cos 4^\circ = -1;$
г) $\sin 17^\circ \cdot \sin 73^\circ - \sin 21^\circ \cdot \cos 13^\circ + \sin 4^\circ \cdot \sin 86^\circ = 0.$
Решение. №770 (с. 218)
Решение 2 (rus). №770 (с. 218)
а) Для доказательства воспользуемся формулой тангенса суммы: $\operatorname{tg}(\alpha + \beta) = \frac{\operatorname{tg}\alpha + \operatorname{tg}\beta}{1 - \operatorname{tg}\alpha \cdot \operatorname{tg}\beta}$.
Пусть $\alpha = 20^{\circ}$ и $\beta = 25^{\circ}$. Тогда их сумма $\alpha + \beta = 20^{\circ} + 25^{\circ} = 45^{\circ}$.
Мы знаем, что $\operatorname{tg}45^{\circ} = 1$.
Подставим значения в формулу:
$\operatorname{tg}(20^{\circ} + 25^{\circ}) = \frac{\operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}}{1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}}$
$1 = \frac{\operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}}{1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}}$
Умножим обе части на знаменатель $(1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ})$:
$1 \cdot (1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}) = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}$
$1 - \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ} = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ}$
Перенесем слагаемое $(-\operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ})$ в правую часть:
$1 = \operatorname{tg}20^{\circ} + \operatorname{tg}25^{\circ} + \operatorname{tg}20^{\circ} \cdot \operatorname{tg}25^{\circ}$
Равенство доказано.
Ответ:
б) Преобразуем левую часть равенства. Воспользуемся формулой приведения $\operatorname{ctg}\alpha = \operatorname{tg}(90^{\circ} - \alpha)$.
$\operatorname{ctg}10^{\circ} = \operatorname{tg}(90^{\circ} - 10^{\circ}) = \operatorname{tg}80^{\circ}$.
Теперь исходное равенство принимает вид:
$\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ} - \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ} = 1$.
Рассмотрим формулу тангенса разности: $\operatorname{tg}(\alpha - \beta) = \frac{\operatorname{tg}\alpha - \operatorname{tg}\beta}{1 + \operatorname{tg}\alpha \cdot \operatorname{tg}\beta}$.
Пусть $\alpha = 80^{\circ}$ и $\beta = 35^{\circ}$. Тогда их разность $\alpha - \beta = 80^{\circ} - 35^{\circ} = 45^{\circ}$.
Мы знаем, что $\operatorname{tg}45^{\circ} = 1$.
$\operatorname{tg}(80^{\circ} - 35^{\circ}) = \frac{\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}}{1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}}$
$1 = \frac{\operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}}{1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}}$
Умножим обе части на знаменатель $(1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ})$:
$1 + \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ} = \operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ}$
Перенесем слагаемое $(\operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ})$ в правую часть:
$1 = \operatorname{tg}80^{\circ} - \operatorname{tg}35^{\circ} - \operatorname{tg}80^{\circ} \cdot \operatorname{tg}35^{\circ}$
Подставив обратно $\operatorname{tg}80^{\circ} = \operatorname{ctg}10^{\circ}$, получаем исходное равенство. Равенство доказано.
Ответ:
в) Преобразуем левую часть равенства, используя формулы приведения $\sin(90^{\circ} - \alpha) = \cos\alpha$:
$\sin 79^{\circ} = \sin(90^{\circ} - 11^{\circ}) = \cos 11^{\circ}$
$\sin 86^{\circ} = \sin(90^{\circ} - 4^{\circ}) = \cos 4^{\circ}$
Подставим эти выражения в исходное равенство:
$\cos 15^{\circ} \cdot \cos 7^{\circ} - \cos 11^{\circ} \cdot \cos 11^{\circ} - \cos 4^{\circ} \cdot \cos 4^{\circ} = \cos 15^{\circ} \cdot \cos 7^{\circ} - \cos^2 11^{\circ} - \cos^2 4^{\circ}$.
Теперь используем формулу произведения косинусов $\cos\alpha\cos\beta = \frac{1}{2}(\cos(\alpha+\beta) + \cos(\alpha-\beta))$ и формулу понижения степени $\cos^2\alpha = \frac{1+\cos(2\alpha)}{2}$.
$\cos 15^{\circ} \cdot \cos 7^{\circ} = \frac{1}{2}(\cos(15^{\circ}+7^{\circ}) + \cos(15^{\circ}-7^{\circ})) = \frac{1}{2}(\cos 22^{\circ} + \cos 8^{\circ})$
$\cos^2 11^{\circ} = \frac{1 + \cos(2 \cdot 11^{\circ})}{2} = \frac{1 + \cos 22^{\circ}}{2}$
$\cos^2 4^{\circ} = \frac{1 + \cos(2 \cdot 4^{\circ})}{2} = \frac{1 + \cos 8^{\circ}}{2}$
Подставим полученные выражения в левую часть:
$\frac{1}{2}(\cos 22^{\circ} + \cos 8^{\circ}) - \frac{1 + \cos 22^{\circ}}{2} - \frac{1 + \cos 8^{\circ}}{2}$
$= \frac{\cos 22^{\circ} + \cos 8^{\circ} - (1 + \cos 22^{\circ}) - (1 + \cos 8^{\circ})}{2}$
$= \frac{\cos 22^{\circ} + \cos 8^{\circ} - 1 - \cos 22^{\circ} - 1 - \cos 8^{\circ}}{2}$
$= \frac{(\cos 22^{\circ} - \cos 22^{\circ}) + (\cos 8^{\circ} - \cos 8^{\circ}) - 1 - 1}{2} = \frac{-2}{2} = -1$
Левая часть равна -1, что совпадает с правой частью. Равенство доказано.
Ответ:
г) Преобразуем левую часть равенства, используя формулы приведения $\sin(90^{\circ} - \alpha) = \cos\alpha$:
$\sin 73^{\circ} = \sin(90^{\circ} - 17^{\circ}) = \cos 17^{\circ}$
$\sin 86^{\circ} = \sin(90^{\circ} - 4^{\circ}) = \cos 4^{\circ}$
Подставим эти выражения в исходное равенство:
$\sin 17^{\circ} \cdot \cos 17^{\circ} - \sin 21^{\circ} \cdot \cos 13^{\circ} + \sin 4^{\circ} \cdot \cos 4^{\circ}$.
Используем формулу синуса двойного угла $\sin(2\alpha) = 2\sin\alpha\cos\alpha$, откуда $\sin\alpha\cos\alpha = \frac{1}{2}\sin(2\alpha)$:
$\sin 17^{\circ} \cdot \cos 17^{\circ} = \frac{1}{2}\sin(2 \cdot 17^{\circ}) = \frac{1}{2}\sin 34^{\circ}$
$\sin 4^{\circ} \cdot \cos 4^{\circ} = \frac{1}{2}\sin(2 \cdot 4^{\circ}) = \frac{1}{2}\sin 8^{\circ}$
Подставим в выражение:
$\frac{1}{2}\sin 34^{\circ} - \sin 21^{\circ} \cdot \cos 13^{\circ} + \frac{1}{2}\sin 8^{\circ} = \frac{1}{2}(\sin 34^{\circ} + \sin 8^{\circ}) - \sin 21^{\circ} \cdot \cos 13^{\circ}$.
Теперь применим формулу суммы синусов $\sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$:
$\sin 34^{\circ} + \sin 8^{\circ} = 2\sin\frac{34^{\circ}+8^{\circ}}{2}\cos\frac{34^{\circ}-8^{\circ}}{2} = 2\sin\frac{42^{\circ}}{2}\cos\frac{26^{\circ}}{2} = 2\sin 21^{\circ}\cos 13^{\circ}$.
Подставим результат обратно в выражение:
$\frac{1}{2}(2\sin 21^{\circ}\cos 13^{\circ}) - \sin 21^{\circ} \cdot \cos 13^{\circ} = \sin 21^{\circ}\cos 13^{\circ} - \sin 21^{\circ}\cos 13^{\circ} = 0$.
Левая часть равна 0, что совпадает с правой частью. Равенство доказано.
Ответ:
Другие задания:
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