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Номер 18, страница 7 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Упражнения для повторения курса алгебры 7—9 классов - номер 18, страница 7.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113608 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-18" "field_display_title" => "18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "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 {#1057 #items: array:1 [ 0 => App\Models\Book {#1056 #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" => 4293 "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 {#1061 #items: array:1 [ 0 => App\Models\Term {#1058 #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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 {#1063 #items: array:1 [ 0 => App\Models\Term {#1060 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1062 #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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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 {#1059 #items: array:2 [ 0 => App\Models\Term {#1074 #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" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "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 {#1075 #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" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1066 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1073 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1076 #items: array:1 [ 0 => App\Models\Term {#1067 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1068 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080 #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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1084 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "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 {#1061} "field_class" => Illuminate\Database\Eloquent\Collection {#1063} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1065} "field_author" => Illuminate\Database\Eloquent\Collection {#1059} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1071} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_country" => Illuminate\Database\Eloquent\Collection {#1070} "field_city" => Illuminate\Database\Eloquent\Collection {#1076} "field_series" => Illuminate\Database\Eloquent\Collection {#1068} "field_umk" => Illuminate\Database\Eloquent\Collection {#1064} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1077} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1079} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080} "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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1081} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1084} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1049 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "4" "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 {#1057} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1049} ] #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 {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1052 #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" => 1290668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1085 #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" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Term {#1086 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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 {#1088 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087} "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 {#1088} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090} "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" => "1113608" "type" => "task" ] "text" => "<p><strong>18.</strong> Упростите выражение:</p><p><strong>а)</strong> $ \frac{4 \cos 4 \alpha}{\operatorname{ctg} 2 \alpha - \operatorname{tg} 2 \alpha} $</p><p><strong>б)</strong> $ \frac{\sin 2 \alpha + \operatorname{tg} 2 \alpha}{1 + \cos 2 \alpha} $</p><p><strong>в)</strong> $ \frac{\operatorname{ctg}^2 2 \alpha - 1}{2 \operatorname{ctg} 2 \alpha} $</p><p><strong>г)</strong> $ \frac{\cos 4 \alpha + \sin^2 2 \alpha}{0.5 \sin 4 \alpha} $</p><p><strong>д)</strong> $ \frac{\sin(2 \pi - 2 \alpha)}{\cos(\alpha + \pi) \cdot \operatorname{ctg}\left(\alpha - \frac{3 \pi}{2}\right)} $</p><p><strong>е)</strong> $ \frac{2 - 2 \sin^2 (\alpha + 0.5 \pi)}{1 - \cos(\alpha - \pi)} - 2 \sin(\alpha + 1.5 \pi) $</p><p><strong>ж)</strong> $ \frac{\cos(\pi + \alpha) \cdot \cos(1.5 - 2 \alpha)}{2 \operatorname{ctg}(\alpha + 0.5 \pi)} $</p><p><strong>з)</strong> $ \frac{2 \sin^2 (\alpha - 2 \pi) - 2}{\cos(\alpha + 1.5 \pi) - 1} - 2 \cos(1.5 \pi + \alpha) $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "18-1.jpg" "alt" => null "width" => "2631" "height" => 1077 "path" => "/media/algebra_10/Abylkasymova-u/0-00/18-1.webp?ts=1753262978" ] ] ] #original: array:7 [ "id" => 1290668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "1113608" "type" => "task" ] "text" => "<p><strong>18.</strong> Упростите выражение:</p><p><strong>а)</strong> $ \frac{4 \cos 4 \alpha}{\operatorname{ctg} 2 \alpha - \operatorname{tg} 2 \alpha} $</p><p><strong>б)</strong> $ \frac{\sin 2 \alpha + \operatorname{tg} 2 \alpha}{1 + \cos 2 \alpha} $</p><p><strong>в)</strong> $ \frac{\operatorname{ctg}^2 2 \alpha - 1}{2 \operatorname{ctg} 2 \alpha} $</p><p><strong>г)</strong> $ \frac{\cos 4 \alpha + \sin^2 2 \alpha}{0.5 \sin 4 \alpha} $</p><p><strong>д)</strong> $ \frac{\sin(2 \pi - 2 \alpha)}{\cos(\alpha + \pi) \cdot \operatorname{ctg}\left(\alpha - \frac{3 \pi}{2}\right)} $</p><p><strong>е)</strong> $ \frac{2 - 2 \sin^2 (\alpha + 0.5 \pi)}{1 - \cos(\alpha - \pi)} - 2 \sin(\alpha + 1.5 \pi) $</p><p><strong>ж)</strong> $ \frac{\cos(\pi + \alpha) \cdot \cos(1.5 - 2 \alpha)}{2 \operatorname{ctg}(\alpha + 0.5 \pi)} $</p><p><strong>з)</strong> $ \frac{2 \sin^2 (\alpha - 2 \pi) - 2}{\cos(\alpha + 1.5 \pi) - 1} - 2 \cos(1.5 \pi + \alpha) $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "18-1.jpg" "alt" => null "width" => "2631" "height" => 1077 "path" => "/media/algebra_10/Abylkasymova-u/0-00/18-1.webp?ts=1753262978" ] ] ] #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 {#1091 #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" => 1291182 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ 0 => App\Models\Term {#1094 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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 {#1096 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095} "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 {#1096} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "1113608" "type" => "task" ] "img" => array:5 [ 0 => array:5 [ "name" => "18-1.jpg" "alt" => null "width" => "1824" "height" => 707 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "18-2.jpg" "alt" => null "width" => "1657" "height" => 777 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "18-3.jpg" "alt" => null "width" => "1971" "height" => 3485 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-3.webp?ts=1753263138" ] 3 => array:5 [ "name" => "18-4.jpg" "alt" => null "width" => "1912" "height" => 3491 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-4.webp?ts=1753263138" ] 4 => array:5 [ "name" => "18-5.jpg" "alt" => null "width" => "1418" "height" => 259 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-5.webp?ts=1753263138" ] ] ] #original: array:6 [ "id" => 1291182 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1101} "task" => array:2 [ "refs" => "1113608" "type" => "task" ] "img" => array:5 [ 0 => array:5 [ "name" => "18-1.jpg" "alt" => null "width" => "1824" "height" => 707 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "18-2.jpg" "alt" => null "width" => "1657" "height" => 777 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "18-3.jpg" "alt" => null "width" => "1971" "height" => 3485 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-3.webp?ts=1753263138" ] 3 => array:5 [ "name" => "18-4.jpg" "alt" => null "width" => "1912" "height" => 3491 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-4.webp?ts=1753263138" ] 4 => array:5 [ "name" => "18-5.jpg" "alt" => null "width" => "1418" "height" => 259 "path" => "/media/algebra_10/Abylkasymova-u/1-00/18-5.webp?ts=1753263138" ] ] ] #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 {#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:6 [ "id" => 1549311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Edition {#1100 #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" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Term {#1102 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103} "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 {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106} "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" => "1113608" "type" => "task" ] "text" => "<p><strong>а)</strong> $\frac{4\cos4\alpha}{\ctg2\alpha - \tg2\alpha}$</p><p>Преобразуем знаменатель, используя определения котангенса и тангенса: $\ctg2\alpha - \tg2\alpha = \frac{\cos2\alpha}{\sin2\alpha} - \frac{\sin2\alpha}{\cos2\alpha}$.</p><p>Приведем к общему знаменателю: $\frac{\cos^22\alpha - \sin^22\alpha}{\sin2\alpha\cos2\alpha}$.</p><p>В числителе получилась формула косинуса двойного угла: $\cos^22\alpha - \sin^22\alpha = \cos(2 \cdot 2\alpha) = \cos4\alpha$.</p><p>В знаменателе используем формулу синуса двойного угла: $\sin2\alpha\cos2\alpha = \frac{1}{2}\sin(2 \cdot 2\alpha) = \frac{1}{2}\sin4\alpha$.</p><p>Таким образом, знаменатель дроби равен: $\frac{\cos4\alpha}{\frac{1}{2}\sin4\alpha} = \frac{2\cos4\alpha}{\sin4\alpha}$.</p><p>Подставим полученное выражение обратно в исходную дробь: $\frac{4\cos4\alpha}{\frac{2\cos4\alpha}{\sin4\alpha}} = 4\cos4\alpha \cdot \frac{\sin4\alpha}{2\cos4\alpha} = 2\sin4\alpha$.</p><p><strong>Ответ:</strong> $2\sin4\alpha$</p><p><strong>б)</strong> $\frac{\sin2\alpha + \tg2\alpha}{1 + \cos2\alpha}$</p><p>Вынесем $\sin2\alpha$ за скобки в числителе: $\sin2\alpha + \tg2\alpha = \sin2\alpha + \frac{\sin2\alpha}{\cos2\alpha} = \sin2\alpha (1 + \frac{1}{\cos2\alpha}) = \sin2\alpha \frac{\cos2\alpha + 1}{\cos2\alpha}$.</p><p>Подставим это в исходное выражение: $\frac{\sin2\alpha \frac{1 + \cos2\alpha}{\cos2\alpha}}{1 + \cos2\alpha}$.</p><p>Сократим $(1 + \cos2\alpha)$: $\frac{\sin2\alpha}{\cos2\alpha} = \tg2\alpha$.</p><p><strong>Ответ:</strong> $\tg2\alpha$</p><p><strong>в)</strong> $\frac{\ctg^22\alpha - 1}{2\ctg2\alpha}$</p><p>Данное выражение представляет собой формулу котангенса двойного угла: $\ctg(2x) = \frac{\ctg^2x - 1}{2\ctg x}$.</p><p>В нашем случае $x = 2\alpha$.</p><p>Следовательно, выражение равно $\ctg(2 \cdot 2\alpha) = \ctg4\alpha$.</p><p><strong>Ответ:</strong> $\ctg4\alpha$</p><p><strong>г)</strong> $\frac{\cos4\alpha + \sin^22\alpha}{0,5\sin4\alpha}$</p><p>Используем формулу косинуса двойного угла для числителя: $\cos4\alpha = \cos^22\alpha - \sin^22\alpha$.</p><p>Числитель становится: $\cos^22\alpha - \sin^22\alpha + \sin^22\alpha = \cos^22\alpha$.</p><p>Преобразуем знаменатель, используя формулу синуса двойного угла: $0,5\sin4\alpha = 0,5 \cdot 2\sin2\alpha\cos2\alpha = \sin2\alpha\cos2\alpha$.</p><p>Получаем дробь: $\frac{\cos^22\alpha}{\sin2\alpha\cos2\alpha}$.</p><p>Сокращаем на $\cos2\alpha$: $\frac{\cos2\alpha}{\sin2\alpha} = \ctg2\alpha$.</p><p><strong>Ответ:</strong> $\ctg2\alpha$</p><p><strong>д)</strong> $\frac{\sin(2\pi - 2\alpha)}{\cos(\alpha + \pi) \cdot \ctg(\alpha - \frac{3\pi}{2})}$</p><p>Применим формулы приведения:</p><p>$\sin(2\pi - 2\alpha) = -\sin(2\alpha)$</p><p>$\cos(\alpha + \pi) = -\cos\alpha$</p><p>$\ctg(\alpha - \frac{3\pi}{2}) = \ctg(-(\frac{3\pi}{2} - \alpha)) = -\ctg(\frac{3\pi}{2} - \alpha) = -\tg\alpha$</p><p>Подставим упрощенные выражения в дробь: $\frac{-\sin(2\alpha)}{(-\cos\alpha) \cdot (-\tg\alpha)} = \frac{-\sin(2\alpha)}{\cos\alpha \cdot \tg\alpha}$.</p><p>Так как $\tg\alpha = \frac{\sin\alpha}{\cos\alpha}$, знаменатель равен $\cos\alpha \cdot \frac{\sin\alpha}{\cos\alpha} = \sin\alpha$.</p><p>Дробь принимает вид: $\frac{-\sin(2\alpha)}{\sin\alpha}$.</p><p>Используем формулу синуса двойного угла $\sin(2\alpha) = 2\sin\alpha\cos\alpha$: $\frac{-2\sin\alpha\cos\alpha}{\sin\alpha} = -2\cos\alpha$.</p><p><strong>Ответ:</strong> $-2\cos\alpha$</p><p><strong>е)</strong> $\frac{2 - 2\sin^2(\alpha + 0,5\pi)}{1 - \cos(\alpha - \pi)} - 2\sin(\alpha + 1,5\pi)$</p><p>Упростим первое слагаемое (дробь). Числитель: $2 - 2\sin^2(\alpha + \frac{\pi}{2}) = 2(1 - \sin^2(\alpha + \frac{\pi}{2})) = 2\cos^2(\alpha + \frac{\pi}{2})$.</p><p>По формуле приведения $\cos(\alpha + \frac{\pi}{2}) = -\sin\alpha$, поэтому числитель равен $2(-\sin\alpha)^2 = 2\sin^2\alpha$.</p><p>Знаменатель: $1 - \cos(\alpha - \pi) = 1 - \cos(\pi - \alpha) = 1 - (-\cos\alpha) = 1 + \cos\alpha$.</p><p>Дробь: $\frac{2\sin^2\alpha}{1 + \cos\alpha} = \frac{2(1-\cos^2\alpha)}{1+\cos\alpha} = \frac{2(1-\cos\alpha)(1+\cos\alpha)}{1+\cos\alpha} = 2(1-\cos\alpha)$.</p><p>Упростим второе слагаемое: $-2\sin(\alpha + 1,5\pi) = -2\sin(\alpha + \frac{3\pi}{2})$.</p><p>По формуле приведения $\sin(\alpha + \frac{3\pi}{2}) = -\cos\alpha$, поэтому второе слагаемое равно $-2(-\cos\alpha) = 2\cos\alpha$.</p><p>Сложим результаты: $2(1-\cos\alpha) + 2\cos\alpha = 2 - 2\cos\alpha + 2\cos\alpha = 2$.</p><p><strong>Ответ:</strong> $2$</p><p><strong>ж)</strong> $\frac{\cos(\pi + \alpha) \cdot \cos(1,5\pi - 2\alpha)}{2\ctg(\alpha + 0,5\pi)}$</p><p>Применим формулы приведения к каждому множителю.</p><p>$\cos(\pi + \alpha) = -\cos\alpha$.</p><p>$\cos(1,5\pi - 2\alpha) = \cos(\frac{3\pi}{2} - 2\alpha) = -\sin(2\alpha)$.</p><p>$\ctg(\alpha + 0,5\pi) = \ctg(\alpha + \frac{\pi}{2}) = -\tg\alpha$.</p><p>Подставляем в выражение: $\frac{(-\cos\alpha) \cdot (-\sin(2\alpha))}{2(-\tg\alpha)} = \frac{\cos\alpha \cdot \sin(2\alpha)}{-2\tg\alpha}$.</p><p>Используем формулу $\sin(2\alpha) = 2\sin\alpha\cos\alpha$: $\frac{\cos\alpha \cdot (2\sin\alpha\cos\alpha)}{-2\frac{\sin\alpha}{\cos\alpha}} = \frac{2\sin\alpha\cos^2\alpha}{-2\frac{\sin\alpha}{\cos\alpha}}$.</p><p>Сокращаем $2\sin\alpha$: $\frac{\cos^2\alpha}{-\frac{1}{\cos\alpha}} = -\cos^2\alpha \cdot \cos\alpha = -\cos^3\alpha$.</p><p><strong>Ответ:</strong> $-\cos^3\alpha$</p><p><strong>з)</strong> $\frac{2\sin^2(\alpha - 2\pi) - 2}{\cos(\alpha + 1,5\pi) - 1} - 2\cos(1,5\pi + \alpha)$</p><p>Упростим первое слагаемое (дробь). Используем периодичность синуса $\sin(\alpha - 2\pi) = \sin\alpha$.</p><p>Числитель: $2\sin^2\alpha - 2 = -2(1 - \sin^2\alpha) = -2\cos^2\alpha$.</p><p>Знаменатель: $\cos(\alpha + 1,5\pi) - 1 = \cos(\alpha + \frac{3\pi}{2}) - 1$. По формуле приведения $\cos(\alpha + \frac{3\pi}{2}) = \sin\alpha$. Знаменатель равен $\sin\alpha - 1$.</p><p>Дробь: $\frac{-2\cos^2\alpha}{\sin\alpha - 1}$. Используем основное тригонометрическое тождество $\cos^2\alpha = 1 - \sin^2\alpha = (1-\sin\alpha)(1+\sin\alpha)$.</p><p>$\frac{-2(1-\sin\alpha)(1+\sin\alpha)}{\sin\alpha - 1} = \frac{2(\sin\alpha - 1)(1+\sin\alpha)}{\sin\alpha - 1} = 2(1+\sin\alpha)$.</p><p>Упростим второе слагаемое: $-2\cos(1,5\pi + \alpha) = -2\cos(\frac{3\pi}{2} + \alpha)$. По формуле приведения $\cos(\frac{3\pi}{2} + \alpha) = \sin\alpha$.</p><p>Второе слагаемое равно $-2\sin\alpha$.</p><p>Соберем все вместе: $2(1+\sin\alpha) - 2\sin\alpha = 2 + 2\sin\alpha - 2\sin\alpha = 2$.</p><p><strong>Ответ:</strong> $2$</p>" ] #original: array:6 [ "id" => 1549311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109} "task" => array:2 [ "refs" => "1113608" "type" => "task" ] "text" => "<p><strong>а)</strong> $\frac{4\cos4\alpha}{\ctg2\alpha - \tg2\alpha}$</p><p>Преобразуем знаменатель, используя определения котангенса и тангенса: $\ctg2\alpha - \tg2\alpha = \frac{\cos2\alpha}{\sin2\alpha} - \frac{\sin2\alpha}{\cos2\alpha}$.</p><p>Приведем к общему знаменателю: $\frac{\cos^22\alpha - \sin^22\alpha}{\sin2\alpha\cos2\alpha}$.</p><p>В числителе получилась формула косинуса двойного угла: $\cos^22\alpha - \sin^22\alpha = \cos(2 \cdot 2\alpha) = \cos4\alpha$.</p><p>В знаменателе используем формулу синуса двойного угла: $\sin2\alpha\cos2\alpha = \frac{1}{2}\sin(2 \cdot 2\alpha) = \frac{1}{2}\sin4\alpha$.</p><p>Таким образом, знаменатель дроби равен: $\frac{\cos4\alpha}{\frac{1}{2}\sin4\alpha} = \frac{2\cos4\alpha}{\sin4\alpha}$.</p><p>Подставим полученное выражение обратно в исходную дробь: $\frac{4\cos4\alpha}{\frac{2\cos4\alpha}{\sin4\alpha}} = 4\cos4\alpha \cdot \frac{\sin4\alpha}{2\cos4\alpha} = 2\sin4\alpha$.</p><p><strong>Ответ:</strong> $2\sin4\alpha$</p><p><strong>б)</strong> $\frac{\sin2\alpha + \tg2\alpha}{1 + \cos2\alpha}$</p><p>Вынесем $\sin2\alpha$ за скобки в числителе: $\sin2\alpha + \tg2\alpha = \sin2\alpha + \frac{\sin2\alpha}{\cos2\alpha} = \sin2\alpha (1 + \frac{1}{\cos2\alpha}) = \sin2\alpha \frac{\cos2\alpha + 1}{\cos2\alpha}$.</p><p>Подставим это в исходное выражение: $\frac{\sin2\alpha \frac{1 + \cos2\alpha}{\cos2\alpha}}{1 + \cos2\alpha}$.</p><p>Сократим $(1 + \cos2\alpha)$: $\frac{\sin2\alpha}{\cos2\alpha} = \tg2\alpha$.</p><p><strong>Ответ:</strong> $\tg2\alpha$</p><p><strong>в)</strong> $\frac{\ctg^22\alpha - 1}{2\ctg2\alpha}$</p><p>Данное выражение представляет собой формулу котангенса двойного угла: $\ctg(2x) = \frac{\ctg^2x - 1}{2\ctg x}$.</p><p>В нашем случае $x = 2\alpha$.</p><p>Следовательно, выражение равно $\ctg(2 \cdot 2\alpha) = \ctg4\alpha$.</p><p><strong>Ответ:</strong> $\ctg4\alpha$</p><p><strong>г)</strong> $\frac{\cos4\alpha + \sin^22\alpha}{0,5\sin4\alpha}$</p><p>Используем формулу косинуса двойного угла для числителя: $\cos4\alpha = \cos^22\alpha - \sin^22\alpha$.</p><p>Числитель становится: $\cos^22\alpha - \sin^22\alpha + \sin^22\alpha = \cos^22\alpha$.</p><p>Преобразуем знаменатель, используя формулу синуса двойного угла: $0,5\sin4\alpha = 0,5 \cdot 2\sin2\alpha\cos2\alpha = \sin2\alpha\cos2\alpha$.</p><p>Получаем дробь: $\frac{\cos^22\alpha}{\sin2\alpha\cos2\alpha}$.</p><p>Сокращаем на $\cos2\alpha$: $\frac{\cos2\alpha}{\sin2\alpha} = \ctg2\alpha$.</p><p><strong>Ответ:</strong> $\ctg2\alpha$</p><p><strong>д)</strong> $\frac{\sin(2\pi - 2\alpha)}{\cos(\alpha + \pi) \cdot \ctg(\alpha - \frac{3\pi}{2})}$</p><p>Применим формулы приведения:</p><p>$\sin(2\pi - 2\alpha) = -\sin(2\alpha)$</p><p>$\cos(\alpha + \pi) = -\cos\alpha$</p><p>$\ctg(\alpha - \frac{3\pi}{2}) = \ctg(-(\frac{3\pi}{2} - \alpha)) = -\ctg(\frac{3\pi}{2} - \alpha) = -\tg\alpha$</p><p>Подставим упрощенные выражения в дробь: $\frac{-\sin(2\alpha)}{(-\cos\alpha) \cdot (-\tg\alpha)} = \frac{-\sin(2\alpha)}{\cos\alpha \cdot \tg\alpha}$.</p><p>Так как $\tg\alpha = \frac{\sin\alpha}{\cos\alpha}$, знаменатель равен $\cos\alpha \cdot \frac{\sin\alpha}{\cos\alpha} = \sin\alpha$.</p><p>Дробь принимает вид: $\frac{-\sin(2\alpha)}{\sin\alpha}$.</p><p>Используем формулу синуса двойного угла $\sin(2\alpha) = 2\sin\alpha\cos\alpha$: $\frac{-2\sin\alpha\cos\alpha}{\sin\alpha} = -2\cos\alpha$.</p><p><strong>Ответ:</strong> $-2\cos\alpha$</p><p><strong>е)</strong> $\frac{2 - 2\sin^2(\alpha + 0,5\pi)}{1 - \cos(\alpha - \pi)} - 2\sin(\alpha + 1,5\pi)$</p><p>Упростим первое слагаемое (дробь). Числитель: $2 - 2\sin^2(\alpha + \frac{\pi}{2}) = 2(1 - \sin^2(\alpha + \frac{\pi}{2})) = 2\cos^2(\alpha + \frac{\pi}{2})$.</p><p>По формуле приведения $\cos(\alpha + \frac{\pi}{2}) = -\sin\alpha$, поэтому числитель равен $2(-\sin\alpha)^2 = 2\sin^2\alpha$.</p><p>Знаменатель: $1 - \cos(\alpha - \pi) = 1 - \cos(\pi - \alpha) = 1 - (-\cos\alpha) = 1 + \cos\alpha$.</p><p>Дробь: $\frac{2\sin^2\alpha}{1 + \cos\alpha} = \frac{2(1-\cos^2\alpha)}{1+\cos\alpha} = \frac{2(1-\cos\alpha)(1+\cos\alpha)}{1+\cos\alpha} = 2(1-\cos\alpha)$.</p><p>Упростим второе слагаемое: $-2\sin(\alpha + 1,5\pi) = -2\sin(\alpha + \frac{3\pi}{2})$.</p><p>По формуле приведения $\sin(\alpha + \frac{3\pi}{2}) = -\cos\alpha$, поэтому второе слагаемое равно $-2(-\cos\alpha) = 2\cos\alpha$.</p><p>Сложим результаты: $2(1-\cos\alpha) + 2\cos\alpha = 2 - 2\cos\alpha + 2\cos\alpha = 2$.</p><p><strong>Ответ:</strong> $2$</p><p><strong>ж)</strong> $\frac{\cos(\pi + \alpha) \cdot \cos(1,5\pi - 2\alpha)}{2\ctg(\alpha + 0,5\pi)}$</p><p>Применим формулы приведения к каждому множителю.</p><p>$\cos(\pi + \alpha) = -\cos\alpha$.</p><p>$\cos(1,5\pi - 2\alpha) = \cos(\frac{3\pi}{2} - 2\alpha) = -\sin(2\alpha)$.</p><p>$\ctg(\alpha + 0,5\pi) = \ctg(\alpha + \frac{\pi}{2}) = -\tg\alpha$.</p><p>Подставляем в выражение: $\frac{(-\cos\alpha) \cdot (-\sin(2\alpha))}{2(-\tg\alpha)} = \frac{\cos\alpha \cdot \sin(2\alpha)}{-2\tg\alpha}$.</p><p>Используем формулу $\sin(2\alpha) = 2\sin\alpha\cos\alpha$: $\frac{\cos\alpha \cdot (2\sin\alpha\cos\alpha)}{-2\frac{\sin\alpha}{\cos\alpha}} = \frac{2\sin\alpha\cos^2\alpha}{-2\frac{\sin\alpha}{\cos\alpha}}$.</p><p>Сокращаем $2\sin\alpha$: $\frac{\cos^2\alpha}{-\frac{1}{\cos\alpha}} = -\cos^2\alpha \cdot \cos\alpha = -\cos^3\alpha$.</p><p><strong>Ответ:</strong> $-\cos^3\alpha$</p><p><strong>з)</strong> $\frac{2\sin^2(\alpha - 2\pi) - 2}{\cos(\alpha + 1,5\pi) - 1} - 2\cos(1,5\pi + \alpha)$</p><p>Упростим первое слагаемое (дробь). Используем периодичность синуса $\sin(\alpha - 2\pi) = \sin\alpha$.</p><p>Числитель: $2\sin^2\alpha - 2 = -2(1 - \sin^2\alpha) = -2\cos^2\alpha$.</p><p>Знаменатель: $\cos(\alpha + 1,5\pi) - 1 = \cos(\alpha + \frac{3\pi}{2}) - 1$. По формуле приведения $\cos(\alpha + \frac{3\pi}{2}) = \sin\alpha$. Знаменатель равен $\sin\alpha - 1$.</p><p>Дробь: $\frac{-2\cos^2\alpha}{\sin\alpha - 1}$. Используем основное тригонометрическое тождество $\cos^2\alpha = 1 - \sin^2\alpha = (1-\sin\alpha)(1+\sin\alpha)$.</p><p>$\frac{-2(1-\sin\alpha)(1+\sin\alpha)}{\sin\alpha - 1} = \frac{2(\sin\alpha - 1)(1+\sin\alpha)}{\sin\alpha - 1} = 2(1+\sin\alpha)$.</p><p>Упростим второе слагаемое: $-2\cos(1,5\pi + \alpha) = -2\cos(\frac{3\pi}{2} + \alpha)$. По формуле приведения $\cos(\frac{3\pi}{2} + \alpha) = \sin\alpha$.</p><p>Второе слагаемое равно $-2\sin\alpha$.</p><p>Соберем все вместе: $2(1+\sin\alpha) - 2\sin\alpha = 2 + 2\sin\alpha - 2\sin\alpha = 2$.</p><p><strong>Ответ:</strong> $2$</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" => "1113609" "type" => "task" ] "previous" => array:2 [ "refs" => "1113607" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\BookPage {#1040 #connection: "mysql" #table: "book_pages" #primaryKey: "id" 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+wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113606 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => "7" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-16" "field_display_title" => "16" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1193 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1194 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:3 [ 0 => App\Models\Element {#1208 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1210 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1212 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113607" "type" => "task" ] "previous" => array:2 [ "refs" => "1113605" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098343" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113606 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => "7" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-16" "field_display_title" => "16" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1193} "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 {#1194} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1195} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1196} "content" => Illuminate\Database\Eloquent\Collection {#1199} "next" => array:2 [ "refs" => "1113607" "type" => "task" ] "previous" => array:2 [ "refs" => "1113605" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1205} "page" => array:2 [ "refs" => "1098343" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1198 #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" => 1113607 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-17" "field_display_title" => "17" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1204 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:3 [ 0 => App\Models\Element {#1222 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1224 #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 [ …6] #original: array:6 [ …6] #changes: [] …15 } 2 => App\Models\Element {#1226 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113608" "type" => "task" ] "previous" => array:2 [ "refs" => "1113606" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113607 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-17" "field_display_title" => "17" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197} "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 {#1204} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1206} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1202} "content" => Illuminate\Database\Eloquent\Collection {#1201} "next" => array:2 [ "refs" => "1113608" "type" => "task" ] "previous" => array:2 [ "refs" => "1113606" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1219} "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1203 #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" => 1113608 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-18" "field_display_title" => "18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1218 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:3 [ 0 => App\Models\Element {#1052} 1 => App\Models\Element {#1091} 2 => App\Models\Element {#1099} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113609" "type" => "task" ] "previous" => array:2 [ "refs" => "1113607" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113608 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-18" "field_display_title" => "18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200} "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 {#1218} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1220} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1216} "content" => Illuminate\Database\Eloquent\Collection {#1214} "next" => array:2 [ "refs" => "1113609" "type" => "task" ] "previous" => array:2 [ "refs" => "1113607" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1217} "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1215 #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" => 1113609 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-19" "field_display_title" => "19" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1229 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:3 [ 0 => App\Models\Element {#1243 …30} 1 => App\Models\Element {#1245 …30} 2 => App\Models\Element {#1247 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113610" "type" => "task" ] "previous" => array:2 [ "refs" => "1113608" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113609 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-19" "field_display_title" => "19" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228} "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 {#1229} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1230} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1231} "content" => Illuminate\Database\Eloquent\Collection {#1234} "next" => array:2 [ "refs" => "1113610" "type" => "task" ] "previous" => array:2 [ "refs" => "1113608" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1240} "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1233 #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" => 1113610 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-20" "field_display_title" => "20" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1239 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:3 [ 0 => App\Models\Element {#1257 …30} 1 => App\Models\Element {#1259 …30} 2 => App\Models\Element {#1261 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113611" "type" => "task" ] "previous" => array:2 [ "refs" => "1113609" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113610 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-20" "field_display_title" => "20" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1232} "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 {#1239} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1241} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1237} "content" => Illuminate\Database\Eloquent\Collection {#1236} "next" => array:2 [ "refs" => "1113611" "type" => "task" ] "previous" => array:2 [ "refs" => "1113609" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1254} "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1238 #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" => 1113611 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-21" "field_display_title" => "21" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1253 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:3 [ 0 => App\Models\Element {#1271 …30} 1 => App\Models\Element {#1273 …30} 2 => App\Models\Element {#1275 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113612" "type" => "task" ] "previous" => array:2 [ "refs" => "1113610" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113611 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-21" "field_display_title" => "21" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1235} "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 {#1253} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1251} "content" => Illuminate\Database\Eloquent\Collection {#1250} "next" => array:2 [ "refs" => "1113612" "type" => "task" ] "previous" => array:2 [ "refs" => "1113610" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1268} "page" => array:2 [ "refs" => "1098344" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1252 #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" => 1113612 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "7" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-22" "field_display_title" => "22" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1267 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1265 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1264 #items: array:3 [ 0 => App\Models\Element 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№18 (с. 7)
Условие. №18 (с. 7)
скриншот условия
18. Упростите выражение:
а) $ \frac{4 \cos 4 \alpha}{\operatorname{ctg} 2 \alpha - \operatorname{tg} 2 \alpha} $
б) $ \frac{\sin 2 \alpha + \operatorname{tg} 2 \alpha}{1 + \cos 2 \alpha} $
в) $ \frac{\operatorname{ctg}^2 2 \alpha - 1}{2 \operatorname{ctg} 2 \alpha} $
г) $ \frac{\cos 4 \alpha + \sin^2 2 \alpha}{0.5 \sin 4 \alpha} $
д) $ \frac{\sin(2 \pi - 2 \alpha)}{\cos(\alpha + \pi) \cdot \operatorname{ctg}\left(\alpha - \frac{3 \pi}{2}\right)} $
е) $ \frac{2 - 2 \sin^2 (\alpha + 0.5 \pi)}{1 - \cos(\alpha - \pi)} - 2 \sin(\alpha + 1.5 \pi) $
ж) $ \frac{\cos(\pi + \alpha) \cdot \cos(1.5 - 2 \alpha)}{2 \operatorname{ctg}(\alpha + 0.5 \pi)} $
з) $ \frac{2 \sin^2 (\alpha - 2 \pi) - 2}{\cos(\alpha + 1.5 \pi) - 1} - 2 \cos(1.5 \pi + \alpha) $
Решение. №18 (с. 7)
Решение 2. №18 (с. 7)
а) $\frac{4\cos4\alpha}{\ctg2\alpha - \tg2\alpha}$
Преобразуем знаменатель, используя определения котангенса и тангенса: $\ctg2\alpha - \tg2\alpha = \frac{\cos2\alpha}{\sin2\alpha} - \frac{\sin2\alpha}{\cos2\alpha}$.
Приведем к общему знаменателю: $\frac{\cos^22\alpha - \sin^22\alpha}{\sin2\alpha\cos2\alpha}$.
В числителе получилась формула косинуса двойного угла: $\cos^22\alpha - \sin^22\alpha = \cos(2 \cdot 2\alpha) = \cos4\alpha$.
В знаменателе используем формулу синуса двойного угла: $\sin2\alpha\cos2\alpha = \frac{1}{2}\sin(2 \cdot 2\alpha) = \frac{1}{2}\sin4\alpha$.
Таким образом, знаменатель дроби равен: $\frac{\cos4\alpha}{\frac{1}{2}\sin4\alpha} = \frac{2\cos4\alpha}{\sin4\alpha}$.
Подставим полученное выражение обратно в исходную дробь: $\frac{4\cos4\alpha}{\frac{2\cos4\alpha}{\sin4\alpha}} = 4\cos4\alpha \cdot \frac{\sin4\alpha}{2\cos4\alpha} = 2\sin4\alpha$.
Ответ: $2\sin4\alpha$
б) $\frac{\sin2\alpha + \tg2\alpha}{1 + \cos2\alpha}$
Вынесем $\sin2\alpha$ за скобки в числителе: $\sin2\alpha + \tg2\alpha = \sin2\alpha + \frac{\sin2\alpha}{\cos2\alpha} = \sin2\alpha (1 + \frac{1}{\cos2\alpha}) = \sin2\alpha \frac{\cos2\alpha + 1}{\cos2\alpha}$.
Подставим это в исходное выражение: $\frac{\sin2\alpha \frac{1 + \cos2\alpha}{\cos2\alpha}}{1 + \cos2\alpha}$.
Сократим $(1 + \cos2\alpha)$: $\frac{\sin2\alpha}{\cos2\alpha} = \tg2\alpha$.
Ответ: $\tg2\alpha$
в) $\frac{\ctg^22\alpha - 1}{2\ctg2\alpha}$
Данное выражение представляет собой формулу котангенса двойного угла: $\ctg(2x) = \frac{\ctg^2x - 1}{2\ctg x}$.
В нашем случае $x = 2\alpha$.
Следовательно, выражение равно $\ctg(2 \cdot 2\alpha) = \ctg4\alpha$.
Ответ: $\ctg4\alpha$
г) $\frac{\cos4\alpha + \sin^22\alpha}{0,5\sin4\alpha}$
Используем формулу косинуса двойного угла для числителя: $\cos4\alpha = \cos^22\alpha - \sin^22\alpha$.
Числитель становится: $\cos^22\alpha - \sin^22\alpha + \sin^22\alpha = \cos^22\alpha$.
Преобразуем знаменатель, используя формулу синуса двойного угла: $0,5\sin4\alpha = 0,5 \cdot 2\sin2\alpha\cos2\alpha = \sin2\alpha\cos2\alpha$.
Получаем дробь: $\frac{\cos^22\alpha}{\sin2\alpha\cos2\alpha}$.
Сокращаем на $\cos2\alpha$: $\frac{\cos2\alpha}{\sin2\alpha} = \ctg2\alpha$.
Ответ: $\ctg2\alpha$
д) $\frac{\sin(2\pi - 2\alpha)}{\cos(\alpha + \pi) \cdot \ctg(\alpha - \frac{3\pi}{2})}$
Применим формулы приведения:
$\sin(2\pi - 2\alpha) = -\sin(2\alpha)$
$\cos(\alpha + \pi) = -\cos\alpha$
$\ctg(\alpha - \frac{3\pi}{2}) = \ctg(-(\frac{3\pi}{2} - \alpha)) = -\ctg(\frac{3\pi}{2} - \alpha) = -\tg\alpha$
Подставим упрощенные выражения в дробь: $\frac{-\sin(2\alpha)}{(-\cos\alpha) \cdot (-\tg\alpha)} = \frac{-\sin(2\alpha)}{\cos\alpha \cdot \tg\alpha}$.
Так как $\tg\alpha = \frac{\sin\alpha}{\cos\alpha}$, знаменатель равен $\cos\alpha \cdot \frac{\sin\alpha}{\cos\alpha} = \sin\alpha$.
Дробь принимает вид: $\frac{-\sin(2\alpha)}{\sin\alpha}$.
Используем формулу синуса двойного угла $\sin(2\alpha) = 2\sin\alpha\cos\alpha$: $\frac{-2\sin\alpha\cos\alpha}{\sin\alpha} = -2\cos\alpha$.
Ответ: $-2\cos\alpha$
е) $\frac{2 - 2\sin^2(\alpha + 0,5\pi)}{1 - \cos(\alpha - \pi)} - 2\sin(\alpha + 1,5\pi)$
Упростим первое слагаемое (дробь). Числитель: $2 - 2\sin^2(\alpha + \frac{\pi}{2}) = 2(1 - \sin^2(\alpha + \frac{\pi}{2})) = 2\cos^2(\alpha + \frac{\pi}{2})$.
По формуле приведения $\cos(\alpha + \frac{\pi}{2}) = -\sin\alpha$, поэтому числитель равен $2(-\sin\alpha)^2 = 2\sin^2\alpha$.
Знаменатель: $1 - \cos(\alpha - \pi) = 1 - \cos(\pi - \alpha) = 1 - (-\cos\alpha) = 1 + \cos\alpha$.
Дробь: $\frac{2\sin^2\alpha}{1 + \cos\alpha} = \frac{2(1-\cos^2\alpha)}{1+\cos\alpha} = \frac{2(1-\cos\alpha)(1+\cos\alpha)}{1+\cos\alpha} = 2(1-\cos\alpha)$.
Упростим второе слагаемое: $-2\sin(\alpha + 1,5\pi) = -2\sin(\alpha + \frac{3\pi}{2})$.
По формуле приведения $\sin(\alpha + \frac{3\pi}{2}) = -\cos\alpha$, поэтому второе слагаемое равно $-2(-\cos\alpha) = 2\cos\alpha$.
Сложим результаты: $2(1-\cos\alpha) + 2\cos\alpha = 2 - 2\cos\alpha + 2\cos\alpha = 2$.
Ответ: $2$
ж) $\frac{\cos(\pi + \alpha) \cdot \cos(1,5\pi - 2\alpha)}{2\ctg(\alpha + 0,5\pi)}$
Применим формулы приведения к каждому множителю.
$\cos(\pi + \alpha) = -\cos\alpha$.
$\cos(1,5\pi - 2\alpha) = \cos(\frac{3\pi}{2} - 2\alpha) = -\sin(2\alpha)$.
$\ctg(\alpha + 0,5\pi) = \ctg(\alpha + \frac{\pi}{2}) = -\tg\alpha$.
Подставляем в выражение: $\frac{(-\cos\alpha) \cdot (-\sin(2\alpha))}{2(-\tg\alpha)} = \frac{\cos\alpha \cdot \sin(2\alpha)}{-2\tg\alpha}$.
Используем формулу $\sin(2\alpha) = 2\sin\alpha\cos\alpha$: $\frac{\cos\alpha \cdot (2\sin\alpha\cos\alpha)}{-2\frac{\sin\alpha}{\cos\alpha}} = \frac{2\sin\alpha\cos^2\alpha}{-2\frac{\sin\alpha}{\cos\alpha}}$.
Сокращаем $2\sin\alpha$: $\frac{\cos^2\alpha}{-\frac{1}{\cos\alpha}} = -\cos^2\alpha \cdot \cos\alpha = -\cos^3\alpha$.
Ответ: $-\cos^3\alpha$
з) $\frac{2\sin^2(\alpha - 2\pi) - 2}{\cos(\alpha + 1,5\pi) - 1} - 2\cos(1,5\pi + \alpha)$
Упростим первое слагаемое (дробь). Используем периодичность синуса $\sin(\alpha - 2\pi) = \sin\alpha$.
Числитель: $2\sin^2\alpha - 2 = -2(1 - \sin^2\alpha) = -2\cos^2\alpha$.
Знаменатель: $\cos(\alpha + 1,5\pi) - 1 = \cos(\alpha + \frac{3\pi}{2}) - 1$. По формуле приведения $\cos(\alpha + \frac{3\pi}{2}) = \sin\alpha$. Знаменатель равен $\sin\alpha - 1$.
Дробь: $\frac{-2\cos^2\alpha}{\sin\alpha - 1}$. Используем основное тригонометрическое тождество $\cos^2\alpha = 1 - \sin^2\alpha = (1-\sin\alpha)(1+\sin\alpha)$.
$\frac{-2(1-\sin\alpha)(1+\sin\alpha)}{\sin\alpha - 1} = \frac{2(\sin\alpha - 1)(1+\sin\alpha)}{\sin\alpha - 1} = 2(1+\sin\alpha)$.
Упростим второе слагаемое: $-2\cos(1,5\pi + \alpha) = -2\cos(\frac{3\pi}{2} + \alpha)$. По формуле приведения $\cos(\frac{3\pi}{2} + \alpha) = \sin\alpha$.
Второе слагаемое равно $-2\sin\alpha$.
Соберем все вместе: $2(1+\sin\alpha) - 2\sin\alpha = 2 + 2\sin\alpha - 2\sin\alpha = 2$.
Ответ: $2$
Другие задания:
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