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Номер 645, страница 187 - 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 645, страница 187.
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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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "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 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=> 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 } "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" => 1107666 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. 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{#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] ] #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} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "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. 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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" => 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 {#1063} "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 {#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" => 1277000 "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" => "1108381" "type" => "task" ] "text" => "<p><strong>645.</strong> Докажите, что при всех допустимых значениях $\alpha$ принимает одно и то же значение выражение:</p><p><strong>а)</strong> $-2(\sin \alpha + \cos \alpha)^2 + 4\operatorname{tg} \alpha \cdot \cos^2 \alpha;$</p><p><strong>б)</strong> $\frac{1}{2}(\sin^4 \alpha + \cos^4 \alpha) + \operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha;$</p><p><strong>в)</strong> $5 + \frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} - \frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha};$</p><p><strong>г)</strong> $\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} \cdot 4\operatorname{ctg}^2 \alpha + 4\sin^2 \alpha.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "645-1.jpg" "alt" => null "width" => "1545" "height" => 634 "path" => "/media/algebra_09/soltan-u19/0-1/645-1.webp?ts=1752834644" ] ] ] #original: array:7 [ "id" => 1277000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108381" "type" => "task" ] "text" => "<p><strong>645.</strong> Докажите, что при всех допустимых значениях $\alpha$ принимает одно и то же значение выражение:</p><p><strong>а)</strong> $-2(\sin \alpha + \cos \alpha)^2 + 4\operatorname{tg} \alpha \cdot \cos^2 \alpha;$</p><p><strong>б)</strong> $\frac{1}{2}(\sin^4 \alpha + \cos^4 \alpha) + \operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha;$</p><p><strong>в)</strong> $5 + \frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} - \frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha};$</p><p><strong>г)</strong> $\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} \cdot 4\operatorname{ctg}^2 \alpha + 4\sin^2 \alpha.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "645-1.jpg" "alt" => null "width" => "1545" "height" => 634 "path" => "/media/algebra_09/soltan-u19/0-1/645-1.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 {#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" => 1278291 "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" => "1108381" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "645-1.jpg" "alt" => null "width" => "1361" "height" => 1196 "path" => "/media/algebra_09/soltan-u19/1-1/645-1.webp?ts=1752831420" ] 1 => array:5 [ "name" => "645-2.jpg" "alt" => null "width" => "1321" "height" => 1089 "path" => "/media/algebra_09/soltan-u19/1-1/645-2.webp?ts=1752831420" ] ] ] #original: array:6 [ "id" => 1278291 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108381" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "645-1.jpg" "alt" => null "width" => "1361" "height" => 1196 "path" => "/media/algebra_09/soltan-u19/1-1/645-1.webp?ts=1752831420" ] 1 => array:5 [ "name" => "645-2.jpg" "alt" => null "width" => "1321" "height" => 1089 "path" => "/media/algebra_09/soltan-u19/1-1/645-2.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 {#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" => 1554246 "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" => "1108381" "type" => "task" ] "text" => "<p><strong>а)</strong> $-2(\sin \alpha + \cos \alpha)^2 + 4\operatorname{tg} \alpha \cdot \cos^2\alpha$</p><p>Раскроем скобки, используя формулу квадрата суммы $(a+b)^2 = a^2+2ab+b^2$:<br>$(\sin \alpha + \cos \alpha)^2 = \sin^2 \alpha + 2\sin \alpha \cos \alpha + \cos^2 \alpha$<br>Применим основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$:<br>$\sin^2 \alpha + \cos^2 \alpha + 2\sin \alpha \cos \alpha = 1 + 2\sin \alpha \cos \alpha$<br>Подставим это в первую часть выражения:<br>$-2(1 + 2\sin \alpha \cos \alpha) = -2 - 4\sin \alpha \cos \alpha$<br>Теперь преобразуем вторую часть выражения, используя определение тангенса $\operatorname{tg} \alpha = \frac{\sin \alpha}{\cos \alpha}$ (при условии, что $\cos \alpha \neq 0$):<br>$4\operatorname{tg} \alpha \cdot \cos^2\alpha = 4 \cdot \frac{\sin \alpha}{\cos \alpha} \cdot \cos^2\alpha = 4\sin \alpha \cos \alpha$<br>Теперь сложим обе преобразованные части:<br>$(-2 - 4\sin \alpha \cos \alpha) + 4\sin \alpha \cos \alpha = -2$<br>Выражение равно -2 при всех допустимых значениях $\alpha$ (где $\cos \alpha \neq 0$).<br><strong>Ответ:</strong> -2.</p><p><strong>б)</strong> $\frac{1}{2}(\sin^4 \alpha + \cos^4 \alpha) + \operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha$</p><p>Преобразуем выражение $\sin^4 \alpha + \cos^4 \alpha$, выделив полный квадрат:<br>$\sin^4 \alpha + \cos^4 \alpha = (\sin^2 \alpha)^2 + (\cos^2 \alpha)^2 = (\sin^2 \alpha + \cos^2 \alpha)^2 - 2\sin^2 \alpha \cos^2 \alpha$<br>Используя основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$, получаем:<br>$1^2 - 2\sin^2 \alpha \cos^2 \alpha = 1 - 2\sin^2 \alpha \cos^2 \alpha$<br>Подставим это в первую часть исходного выражения:<br>$\frac{1}{2}(1 - 2\sin^2 \alpha \cos^2 \alpha) = \frac{1}{2} - \sin^2 \alpha \cos^2 \alpha$<br>Теперь преобразуем вторую часть выражения, используя определение котангенса $\operatorname{ctg} \alpha = \frac{\cos \alpha}{\sin \alpha}$ (при условии, что $\sin \alpha \neq 0$):<br>$\operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha} \cdot \sin^4 \alpha = \cos^2 \alpha \sin^2 \alpha$<br>Сложим обе преобразованные части:<br>$(\frac{1}{2} - \sin^2 \alpha \cos^2 \alpha) + \cos^2 \alpha \sin^2 \alpha = \frac{1}{2}$<br>Выражение равно $\frac{1}{2}$ при всех допустимых значениях $\alpha$ (где $\sin \alpha \neq 0$).<br><strong>Ответ:</strong> $\frac{1}{2}$.</p><p><strong>в)</strong> $5 + \frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} - \frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha}$</p><p>Используем известные тригонометрические тождества: $1 + \operatorname{tg}^2 \alpha = \frac{1}{\cos^2 \alpha}$ и $1 + \operatorname{ctg}^2 \alpha = \frac{1}{\sin^2 \alpha}$. Допустимыми значениями $\alpha$ являются те, для которых существуют $\operatorname{tg} \alpha$ и $\operatorname{ctg} \alpha$, то есть $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$.<br>Преобразуем первую дробь:<br>$\frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} = \frac{\frac{\sin \alpha}{\cos \alpha}}{\frac{1}{\cos^2 \alpha}} = \frac{\sin \alpha}{\cos \alpha} \cdot \cos^2 \alpha = \sin \alpha \cos \alpha$<br>Преобразуем вторую дробь:<br>$\frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha} = \frac{\frac{\cos \alpha}{\sin \alpha}}{\frac{1}{\sin^2 \alpha}} = \frac{\cos \alpha}{\sin \alpha} \cdot \sin^2 \alpha = \cos \alpha \sin \alpha$<br>Подставим преобразованные дроби в исходное выражение:<br>$5 + \sin \alpha \cos \alpha - \cos \alpha \sin \alpha = 5$<br>Выражение равно 5 при всех допустимых значениях $\alpha$.<br><strong>Ответ:</strong> 5.</p><p><strong>г)</strong> $\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} \cdot 4\operatorname{ctg}^2\alpha + 4\sin^2\alpha$</p><p>Преобразуем числитель первой дроби, используя формулу разности квадратов $a^2-b^2=(a-b)(a+b)$:<br>$\sin^4 \alpha - \cos^4 \alpha = (\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha)$<br>Так как $\sin^2 \alpha + \cos^2 \alpha = 1$, числитель равен $\sin^2 \alpha - \cos^2 \alpha$.<br>Преобразуем знаменатель первой дроби, используя определение котангенса (при $\sin \alpha \neq 0$):<br>$1 - \operatorname{ctg}^2 \alpha = 1 - \frac{\cos^2 \alpha}{\sin^2 \alpha} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\sin^2 \alpha}$<br>Теперь преобразуем всю первую дробь (при условии, что $1 - \operatorname{ctg}^2 \alpha \neq 0$, то есть $\operatorname{ctg}^2 \alpha \neq 1$):<br>$\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\frac{\sin^2 \alpha - \cos^2 \alpha}{\sin^2 \alpha}} = \sin^2 \alpha$<br>Подставим полученный результат в исходное выражение:<br>$(\sin^2 \alpha) \cdot 4\operatorname{ctg}^2\alpha + 4\sin^2\alpha$<br>Преобразуем первое слагаемое, используя определение котангенса:<br>$\sin^2 \alpha \cdot 4 \cdot \frac{\cos^2 \alpha}{\sin^2 \alpha} = 4\cos^2 \alpha$<br>Теперь всё выражение принимает вид:<br>$4\cos^2 \alpha + 4\sin^2\alpha = 4(\cos^2 \alpha + \sin^2 \alpha)$<br>Используя основное тригонометрическое тождество, получаем:<br>$4(1) = 4$<br>Выражение равно 4 при всех допустимых значениях $\alpha$ (где $\sin \alpha \neq 0$ и $\operatorname{ctg}^2 \alpha \neq 1$).<br><strong>Ответ:</strong> 4.</p>" ] #original: array:6 [ "id" => 1554246 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108381" "type" => "task" ] "text" => "<p><strong>а)</strong> $-2(\sin \alpha + \cos \alpha)^2 + 4\operatorname{tg} \alpha \cdot \cos^2\alpha$</p><p>Раскроем скобки, используя формулу квадрата суммы $(a+b)^2 = a^2+2ab+b^2$:<br>$(\sin \alpha + \cos \alpha)^2 = \sin^2 \alpha + 2\sin \alpha \cos \alpha + \cos^2 \alpha$<br>Применим основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$:<br>$\sin^2 \alpha + \cos^2 \alpha + 2\sin \alpha \cos \alpha = 1 + 2\sin \alpha \cos \alpha$<br>Подставим это в первую часть выражения:<br>$-2(1 + 2\sin \alpha \cos \alpha) = -2 - 4\sin \alpha \cos \alpha$<br>Теперь преобразуем вторую часть выражения, используя определение тангенса $\operatorname{tg} \alpha = \frac{\sin \alpha}{\cos \alpha}$ (при условии, что $\cos \alpha \neq 0$):<br>$4\operatorname{tg} \alpha \cdot \cos^2\alpha = 4 \cdot \frac{\sin \alpha}{\cos \alpha} \cdot \cos^2\alpha = 4\sin \alpha \cos \alpha$<br>Теперь сложим обе преобразованные части:<br>$(-2 - 4\sin \alpha \cos \alpha) + 4\sin \alpha \cos \alpha = -2$<br>Выражение равно -2 при всех допустимых значениях $\alpha$ (где $\cos \alpha \neq 0$).<br><strong>Ответ:</strong> -2.</p><p><strong>б)</strong> $\frac{1}{2}(\sin^4 \alpha + \cos^4 \alpha) + \operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha$</p><p>Преобразуем выражение $\sin^4 \alpha + \cos^4 \alpha$, выделив полный квадрат:<br>$\sin^4 \alpha + \cos^4 \alpha = (\sin^2 \alpha)^2 + (\cos^2 \alpha)^2 = (\sin^2 \alpha + \cos^2 \alpha)^2 - 2\sin^2 \alpha \cos^2 \alpha$<br>Используя основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$, получаем:<br>$1^2 - 2\sin^2 \alpha \cos^2 \alpha = 1 - 2\sin^2 \alpha \cos^2 \alpha$<br>Подставим это в первую часть исходного выражения:<br>$\frac{1}{2}(1 - 2\sin^2 \alpha \cos^2 \alpha) = \frac{1}{2} - \sin^2 \alpha \cos^2 \alpha$<br>Теперь преобразуем вторую часть выражения, используя определение котангенса $\operatorname{ctg} \alpha = \frac{\cos \alpha}{\sin \alpha}$ (при условии, что $\sin \alpha \neq 0$):<br>$\operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha} \cdot \sin^4 \alpha = \cos^2 \alpha \sin^2 \alpha$<br>Сложим обе преобразованные части:<br>$(\frac{1}{2} - \sin^2 \alpha \cos^2 \alpha) + \cos^2 \alpha \sin^2 \alpha = \frac{1}{2}$<br>Выражение равно $\frac{1}{2}$ при всех допустимых значениях $\alpha$ (где $\sin \alpha \neq 0$).<br><strong>Ответ:</strong> $\frac{1}{2}$.</p><p><strong>в)</strong> $5 + \frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} - \frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha}$</p><p>Используем известные тригонометрические тождества: $1 + \operatorname{tg}^2 \alpha = \frac{1}{\cos^2 \alpha}$ и $1 + \operatorname{ctg}^2 \alpha = \frac{1}{\sin^2 \alpha}$. Допустимыми значениями $\alpha$ являются те, для которых существуют $\operatorname{tg} \alpha$ и $\operatorname{ctg} \alpha$, то есть $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$.<br>Преобразуем первую дробь:<br>$\frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} = \frac{\frac{\sin \alpha}{\cos \alpha}}{\frac{1}{\cos^2 \alpha}} = \frac{\sin \alpha}{\cos \alpha} \cdot \cos^2 \alpha = \sin \alpha \cos \alpha$<br>Преобразуем вторую дробь:<br>$\frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha} = \frac{\frac{\cos \alpha}{\sin \alpha}}{\frac{1}{\sin^2 \alpha}} = \frac{\cos \alpha}{\sin \alpha} \cdot \sin^2 \alpha = \cos \alpha \sin \alpha$<br>Подставим преобразованные дроби в исходное выражение:<br>$5 + \sin \alpha \cos \alpha - \cos \alpha \sin \alpha = 5$<br>Выражение равно 5 при всех допустимых значениях $\alpha$.<br><strong>Ответ:</strong> 5.</p><p><strong>г)</strong> $\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} \cdot 4\operatorname{ctg}^2\alpha + 4\sin^2\alpha$</p><p>Преобразуем числитель первой дроби, используя формулу разности квадратов $a^2-b^2=(a-b)(a+b)$:<br>$\sin^4 \alpha - \cos^4 \alpha = (\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha)$<br>Так как $\sin^2 \alpha + \cos^2 \alpha = 1$, числитель равен $\sin^2 \alpha - \cos^2 \alpha$.<br>Преобразуем знаменатель первой дроби, используя определение котангенса (при $\sin \alpha \neq 0$):<br>$1 - \operatorname{ctg}^2 \alpha = 1 - \frac{\cos^2 \alpha}{\sin^2 \alpha} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\sin^2 \alpha}$<br>Теперь преобразуем всю первую дробь (при условии, что $1 - \operatorname{ctg}^2 \alpha \neq 0$, то есть $\operatorname{ctg}^2 \alpha \neq 1$):<br>$\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\frac{\sin^2 \alpha - \cos^2 \alpha}{\sin^2 \alpha}} = \sin^2 \alpha$<br>Подставим полученный результат в исходное выражение:<br>$(\sin^2 \alpha) \cdot 4\operatorname{ctg}^2\alpha + 4\sin^2\alpha$<br>Преобразуем первое слагаемое, используя определение котангенса:<br>$\sin^2 \alpha \cdot 4 \cdot \frac{\cos^2 \alpha}{\sin^2 \alpha} = 4\cos^2 \alpha$<br>Теперь всё выражение принимает вид:<br>$4\cos^2 \alpha + 4\sin^2\alpha = 4(\cos^2 \alpha + \sin^2 \alpha)$<br>Используя основное тригонометрическое тождество, получаем:<br>$4(1) = 4$<br>Выражение равно 4 при всех допустимых значениях $\alpha$ (где $\sin \alpha \neq 0$ и $\operatorname{ctg}^2 \alpha \neq 1$).<br><strong>Ответ:</strong> 4.</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" => "1108382" "type" => "task" ] "previous" => array:2 [ "refs" => "1108380" "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" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Term {#1126 #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 {#1127 #items: array:1 [ 0 => App\Models\Term {#1128 #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 {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 #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 {#1131 #items: array:3 [ 0 => App\Models\Term {#1132 #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 {#1133 #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 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#touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1138 #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: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 {#1140 #items: array:1 [ 0 => App\Models\Term {#1141 #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 {#1142 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147 #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 {#1148 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151 #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 {#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 {#2001 #items: array:1 [ 0 => App\Models\BookPage {#1155 #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" => 1097929 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-187" "field_display_title" => "187" "field_folder" => "folder1" "field_image_name" => "187" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1160 #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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …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 {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …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 {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …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 {#1188 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Element {#1190 #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" => 1261080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …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" => "1097930" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097928" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:6 [ 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" => 1108379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-643" "field_display_title" => "643" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1330 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-643" "field_display_title" => "643" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1330 …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 {#1359 #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" => 1108380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-644" "field_display_title" => "644" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1360 …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 {#1362 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1363 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1396 …2} "content" => Illuminate\Database\Eloquent\Collection {#1433 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1458 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-644" "field_display_title" => "644" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1360 …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 {#1362 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1363 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1396 …2} "content" => Illuminate\Database\Eloquent\Collection {#1433 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1458 …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 {#1487 #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" => 1108381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-645" "field_display_title" => "645" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1488 …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 {#1490 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1491 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1524 …2} "content" => Illuminate\Database\Eloquent\Collection {#1561 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1586 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-645" "field_display_title" => "645" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1488 …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 {#1490 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1491 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1524 …2} "content" => Illuminate\Database\Eloquent\Collection {#1561 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1586 …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 {#1615 #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" => 1108382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "187" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-646" "field_display_title" => "646" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1616 …2} "field_metatags_title" => null 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№645 (с. 187)
Условие. №645 (с. 187)
скриншот условия
645. Докажите, что при всех допустимых значениях $\alpha$ принимает одно и то же значение выражение:
а) $-2(\sin \alpha + \cos \alpha)^2 + 4\operatorname{tg} \alpha \cdot \cos^2 \alpha;$
б) $\frac{1}{2}(\sin^4 \alpha + \cos^4 \alpha) + \operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha;$
в) $5 + \frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} - \frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha};$
г) $\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} \cdot 4\operatorname{ctg}^2 \alpha + 4\sin^2 \alpha.$
Решение. №645 (с. 187)
Решение 2 (rus). №645 (с. 187)
а) $-2(\sin \alpha + \cos \alpha)^2 + 4\operatorname{tg} \alpha \cdot \cos^2\alpha$
Раскроем скобки, используя формулу квадрата суммы $(a+b)^2 = a^2+2ab+b^2$:
$(\sin \alpha + \cos \alpha)^2 = \sin^2 \alpha + 2\sin \alpha \cos \alpha + \cos^2 \alpha$
Применим основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$:
$\sin^2 \alpha + \cos^2 \alpha + 2\sin \alpha \cos \alpha = 1 + 2\sin \alpha \cos \alpha$
Подставим это в первую часть выражения:
$-2(1 + 2\sin \alpha \cos \alpha) = -2 - 4\sin \alpha \cos \alpha$
Теперь преобразуем вторую часть выражения, используя определение тангенса $\operatorname{tg} \alpha = \frac{\sin \alpha}{\cos \alpha}$ (при условии, что $\cos \alpha \neq 0$):
$4\operatorname{tg} \alpha \cdot \cos^2\alpha = 4 \cdot \frac{\sin \alpha}{\cos \alpha} \cdot \cos^2\alpha = 4\sin \alpha \cos \alpha$
Теперь сложим обе преобразованные части:
$(-2 - 4\sin \alpha \cos \alpha) + 4\sin \alpha \cos \alpha = -2$
Выражение равно -2 при всех допустимых значениях $\alpha$ (где $\cos \alpha \neq 0$).
Ответ: -2.
б) $\frac{1}{2}(\sin^4 \alpha + \cos^4 \alpha) + \operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha$
Преобразуем выражение $\sin^4 \alpha + \cos^4 \alpha$, выделив полный квадрат:
$\sin^4 \alpha + \cos^4 \alpha = (\sin^2 \alpha)^2 + (\cos^2 \alpha)^2 = (\sin^2 \alpha + \cos^2 \alpha)^2 - 2\sin^2 \alpha \cos^2 \alpha$
Используя основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$, получаем:
$1^2 - 2\sin^2 \alpha \cos^2 \alpha = 1 - 2\sin^2 \alpha \cos^2 \alpha$
Подставим это в первую часть исходного выражения:
$\frac{1}{2}(1 - 2\sin^2 \alpha \cos^2 \alpha) = \frac{1}{2} - \sin^2 \alpha \cos^2 \alpha$
Теперь преобразуем вторую часть выражения, используя определение котангенса $\operatorname{ctg} \alpha = \frac{\cos \alpha}{\sin \alpha}$ (при условии, что $\sin \alpha \neq 0$):
$\operatorname{ctg}^2 \alpha \cdot \sin^4 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha} \cdot \sin^4 \alpha = \cos^2 \alpha \sin^2 \alpha$
Сложим обе преобразованные части:
$(\frac{1}{2} - \sin^2 \alpha \cos^2 \alpha) + \cos^2 \alpha \sin^2 \alpha = \frac{1}{2}$
Выражение равно $\frac{1}{2}$ при всех допустимых значениях $\alpha$ (где $\sin \alpha \neq 0$).
Ответ: $\frac{1}{2}$.
в) $5 + \frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} - \frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha}$
Используем известные тригонометрические тождества: $1 + \operatorname{tg}^2 \alpha = \frac{1}{\cos^2 \alpha}$ и $1 + \operatorname{ctg}^2 \alpha = \frac{1}{\sin^2 \alpha}$. Допустимыми значениями $\alpha$ являются те, для которых существуют $\operatorname{tg} \alpha$ и $\operatorname{ctg} \alpha$, то есть $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$.
Преобразуем первую дробь:
$\frac{\operatorname{tg} \alpha}{1 + \operatorname{tg}^2 \alpha} = \frac{\frac{\sin \alpha}{\cos \alpha}}{\frac{1}{\cos^2 \alpha}} = \frac{\sin \alpha}{\cos \alpha} \cdot \cos^2 \alpha = \sin \alpha \cos \alpha$
Преобразуем вторую дробь:
$\frac{\operatorname{ctg} \alpha}{1 + \operatorname{ctg}^2 \alpha} = \frac{\frac{\cos \alpha}{\sin \alpha}}{\frac{1}{\sin^2 \alpha}} = \frac{\cos \alpha}{\sin \alpha} \cdot \sin^2 \alpha = \cos \alpha \sin \alpha$
Подставим преобразованные дроби в исходное выражение:
$5 + \sin \alpha \cos \alpha - \cos \alpha \sin \alpha = 5$
Выражение равно 5 при всех допустимых значениях $\alpha$.
Ответ: 5.
г) $\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} \cdot 4\operatorname{ctg}^2\alpha + 4\sin^2\alpha$
Преобразуем числитель первой дроби, используя формулу разности квадратов $a^2-b^2=(a-b)(a+b)$:
$\sin^4 \alpha - \cos^4 \alpha = (\sin^2 \alpha - \cos^2 \alpha)(\sin^2 \alpha + \cos^2 \alpha)$
Так как $\sin^2 \alpha + \cos^2 \alpha = 1$, числитель равен $\sin^2 \alpha - \cos^2 \alpha$.
Преобразуем знаменатель первой дроби, используя определение котангенса (при $\sin \alpha \neq 0$):
$1 - \operatorname{ctg}^2 \alpha = 1 - \frac{\cos^2 \alpha}{\sin^2 \alpha} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\sin^2 \alpha}$
Теперь преобразуем всю первую дробь (при условии, что $1 - \operatorname{ctg}^2 \alpha \neq 0$, то есть $\operatorname{ctg}^2 \alpha \neq 1$):
$\frac{\sin^4 \alpha - \cos^4 \alpha}{1 - \operatorname{ctg}^2 \alpha} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\frac{\sin^2 \alpha - \cos^2 \alpha}{\sin^2 \alpha}} = \sin^2 \alpha$
Подставим полученный результат в исходное выражение:
$(\sin^2 \alpha) \cdot 4\operatorname{ctg}^2\alpha + 4\sin^2\alpha$
Преобразуем первое слагаемое, используя определение котангенса:
$\sin^2 \alpha \cdot 4 \cdot \frac{\cos^2 \alpha}{\sin^2 \alpha} = 4\cos^2 \alpha$
Теперь всё выражение принимает вид:
$4\cos^2 \alpha + 4\sin^2\alpha = 4(\cos^2 \alpha + \sin^2 \alpha)$
Используя основное тригонометрическое тождество, получаем:
$4(1) = 4$
Выражение равно 4 при всех допустимых значениях $\alpha$ (где $\sin \alpha \neq 0$ и $\operatorname{ctg}^2 \alpha \neq 1$).
Ответ: 4.
Другие задания:
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