1) Can we conclude that the equation cos(α–β) = cos α – cos β is an...

Question
1) Can we conclude that the equation cos(α–β) = cos α – cos β is an identity? Explain.
2) Can sin θ, cos θ, and tan θ all be negative for any angle θ? Explain.
3) Explain, in your own words, how reference angles are used to find the trigonometric functions of an obtuse angle.
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1) Can we conclude that the equation cos(α–β) = cos α – cos β is an identity? Explain.

No, we can ́t. Suffice to show a pair of values of α and β for which the equation is false. Indeed, let α = β = 0 . The left side becomes cos(0 – 0) = cos(0) = 1 while the right side is cos(0) – cos(0) = 1 – 1 = 0. So α = β = 0 is a counter example, hence cos(α – β) = cos α – cos β is not an identity.
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      • Illuminate\Cache\Events\CacheHit (279μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (249μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (151μs)
      • creating: site.layouts.app (686μs)
      • composing: site.layouts.app (20μs)
      • creating: components.canonical (360μs)
      • composing: components.canonical (104μs)
      • creating: components.open-graph (206μs)
      • composing: components.open-graph (77μs)
      • creating: site.headers.header (427μs)
      • composing: site.headers.header (88μs)
      • Illuminate\Cache\Events\CacheHit (2.05ms)
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      • Illuminate\Database\Events\StatementPrepared (850μs)
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      • Illuminate\Cache\Events\KeyWritten (372μs)
      • Illuminate\Cache\Events\CacheHit (339μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (404μs)
      • Illuminate\Cache\Events\CacheHit (351μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (346μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (314μs)
      • creating: components.forms.contact-us (359μs)
      • composing: components.forms.contact-us (197μs)
      • creating: components.forms.get-started (212μs)
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      • creating: components.forms.free-tool-download (167μs)
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      • creating: components.forms.claim-free-worksheet (153μs)
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      • creating: components.forms.tutor-subscription-waitlist (148μs)
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      • creating: components.forms.tutor-subscription-join (141μs)
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      • creating: components.forms.tutor-support (205μs)
      • composing: components.forms.tutor-support (268μs)
      • 313 x Illuminate\Cache\Events\CacheHit (21.21%)
        131ms
        18 x Illuminate\Database\Events\StatementPrepared (6.5%)
        40.19ms
        18 x Illuminate\Database\Events\QueryExecuted (6.27%)
        38.77ms
        9 x Illuminate\Cache\Events\CacheMissed (2.76%)
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        686μs
        1 x Illuminate\Routing\Events\RouteMatched (0.1%)
        589μs
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        427μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
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        8 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.06%)
        362μs
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        205μs
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        197μs
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        167μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        154μs
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        148μs
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        1 x composing: homework.show (0.02%)
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        104μs
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        96μs
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        88μs
        1 x creating: components.footer (0.01%)
        87μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        87μs
        1 x composing: components.breadcrumbs (0.01%)
        85μs
        1 x composing: components.open-graph (0.01%)
        77μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        76μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        74μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        63μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        60μs
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        1 x composing: site.layouts.app (0%)
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      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
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      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
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      App\Http\Controllers\HomeworkLibraryController@show
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      where
      as
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        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '46944' limit 1
        17.25mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
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        • 1: 0
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        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (4)
        1mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (46944)
        3.95mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.2mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 46944 and `subject` = 4 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        4mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 46944
        • 1: 4
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (4)
        1.4mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 46944 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.35mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 46944
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 46944 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.18mstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 46944 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.08mstwenty4_siteHomeworkLibrary.php#260
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        2.17mstwenty4_siteHomeworkLibrary.php#201
        Bindings
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.39mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        6.18mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        3.02mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.46mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.41mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.06mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.05mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.25mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
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      App\Models\HomeworkLibrary\HomeworkLibraryFile
      8HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
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      App\Models\Subject
      2Subject.php
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