{"id":233761,"date":"2025-06-13T11:37:00","date_gmt":"2025-06-13T11:37:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233761"},"modified":"2025-06-13T11:37:03","modified_gmt":"2025-06-13T11:37:03","slug":"review-the-proof-of-cos-a-b","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/review-the-proof-of-cos-a-b\/","title":{"rendered":"Review the proof of\u00a0cos A-B"},"content":{"rendered":"\n

Review the proof of\u00a0cos A-B
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Here is the correct choice and a detailed explanation of the reasoning.<\/p>\n\n\n\n

Correct Answer:<\/strong>
The correct option to complete step 4 of the proof is 2 and 1<\/strong>.<\/p>\n\n\n\n

Explanation<\/h3>\n\n\n\n

The purpose of Step 4 is to simplify the expanded equation from Step 3 by applying the fundamental Pythagorean identity, which states that for any angle \u03b8, cos\u00b2\u03b8 + sin\u00b2\u03b8 = 1<\/strong>. Let’s break down how this is applied to both sides of the equation.<\/p>\n\n\n\n

Step 3:<\/strong>
The equation after expanding the squares in Step 2 is:
cos\u00b2A – 2cosAcosB + cos\u00b2B + sin\u00b2A – 2sinAsinB + sin\u00b2B = cos\u00b2(A – B) – 2cos(A – B) + 1 + sin\u00b2(A – B)<\/p>\n\n\n\n

Deriving Step 4:<\/strong><\/p>\n\n\n\n

    \n
  1. Simplify the Left-Hand Side (LHS):<\/strong>
    We can regroup the terms on the LHS of the equation from Step 3 to pair up the squared sine and cosine terms:
    (cos\u00b2A + sin\u00b2A) + (cos\u00b2B + sin\u00b2B) – 2cosAcosB – 2sinAsinB
    Applying the Pythagorean identity to\u00a0(cos\u00b2A + sin\u00b2A)\u00a0gives\u00a01.
    Applying the Pythagorean identity to\u00a0(cos\u00b2B + sin\u00b2B)\u00a0also gives\u00a01.
    Substituting these values back, the LHS simplifies to:
    1 + 1 – 2cosAcosB – 2sinAsinB
    = 2 – 2cosAcosB – 2sinAsinB
    Therefore, the first blank in Step 4 is\u00a02<\/strong>.<\/li>\n\n\n\n
  2. Simplify the Right-Hand Side (RHS):<\/strong>
    Similarly, we regroup the terms on the RHS of the equation from Step 3:
    (cos\u00b2(A – B) + sin\u00b2(A – B)) – 2cos(A – B) + 1
    Applying the Pythagorean identity to\u00a0(cos\u00b2(A – B) + sin\u00b2(A – B))\u00a0gives\u00a01.
    Substituting this value back, the RHS becomes:
    1 – 2cos(A – B) + 1
    Therefore, the second blank in Step 4 is\u00a01<\/strong>.<\/li>\n<\/ol>\n\n\n\n

    Completed Step 4:<\/strong>
    By filling in the blanks with our derived values, the complete Step 4 is:
    2<\/strong> – 2cosAcosB – 2sinAsinB = 1<\/strong> – 2cos(A – B) + 1<\/p>\n\n\n\n

    This equation simplifies further to 2 – 2cosAcosB – 2sinAsinB = 2 – 2cos(A – B). Subtracting 2 from both sides leads directly to Step 5, confirming that our choice is the correct logical progression in the proof.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n



























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    \"\"<\/figure>\n\n\n\n

    The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

    Review the proof of\u00a0cos A-B The Correct Answer and Explanation is: Here is the correct choice and a detailed explanation of the reasoning. Correct Answer:The correct option to complete step 4 of the proof is 2 and 1. Explanation The purpose of Step 4 is to simplify the expanded equation from Step 3 by applying the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-233761","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233761","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=233761"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233761\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233761"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233761"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233761"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}