{"id":264783,"date":"2025-07-21T16:14:33","date_gmt":"2025-07-21T16:14:33","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=264783"},"modified":"2025-07-21T16:14:36","modified_gmt":"2025-07-21T16:14:36","slug":"fraccos-70circsin-20circ-cos-36circ-csc-54circ","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/21\/fraccos-70circsin-20circ-cos-36circ-csc-54circ\/","title":{"rendered":"\\frac{\\cos 70^\\circ}{\\sin 20^\\circ} + \\cos 36^\\circ \\csc 54^\\circ"},"content":{"rendered":"\n

\\frac{\\cos 70^\\circ}{\\sin 20^\\circ} + \\cos 36^\\circ \\csc 54^\\circ<\/p>\n\n\n\n

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

The expression we are looking to simplify is:cos\u206170\u2218sin\u206120\u2218+cos\u206136\u2218csc\u206154\u2218\\frac{\\cos 70^\\circ}{\\sin 20^\\circ} + \\cos 36^\\circ \\csc 54^\\circsin20\u2218cos70\u2218\u200b+cos36\u2218csc54\u2218<\/p>\n\n\n\n

Step-by-Step Simplification:<\/h3>\n\n\n\n
    \n
  1. First part:<\/strong> cos\u206170\u2218sin\u206120\u2218\\frac{\\cos 70^\\circ}{\\sin 20^\\circ}sin20\u2218cos70\u2218\u200b We know from trigonometric identities that: cos\u206170\u2218=sin\u206120\u2218\\cos 70^\\circ = \\sin 20^\\circcos70\u2218=sin20\u2218 Therefore, the first part of the expression simplifies to: sin\u206120\u2218sin\u206120\u2218=1\\frac{\\sin 20^\\circ}{\\sin 20^\\circ} = 1sin20\u2218sin20\u2218\u200b=1<\/li>\n\n\n\n
  2. Second part:<\/strong> cos\u206136\u2218csc\u206154\u2218\\cos 36^\\circ \\csc 54^\\circcos36\u2218csc54\u2218 We know that: csc\u2061\u03b8=1sin\u2061\u03b8\\csc \\theta = \\frac{1}{\\sin \\theta}csc\u03b8=sin\u03b81\u200b Hence, csc\u206154\u2218=1sin\u206154\u2218\\csc 54^\\circ = \\frac{1}{\\sin 54^\\circ}csc54\u2218=sin54\u22181\u200b Therefore, the second part becomes: cos\u206136\u2218\u00d71sin\u206154\u2218\\cos 36^\\circ \\times \\frac{1}{\\sin 54^\\circ}cos36\u2218\u00d7sin54\u22181\u200b We also know that: sin\u206154\u2218=cos\u206136\u2218\\sin 54^\\circ = \\cos 36^\\circsin54\u2218=cos36\u2218 Thus, the second part simplifies to: cos\u206136\u2218\u00d71cos\u206136\u2218=1\\cos 36^\\circ \\times \\frac{1}{\\cos 36^\\circ} = 1cos36\u2218\u00d7cos36\u22181\u200b=1<\/li>\n<\/ol>\n\n\n\n

    Final Answer:<\/h3>\n\n\n\n

    So, the entire expression simplifies as follows:1+1=21 + 1 = 21+1=2<\/p>\n\n\n\n

    Thus, the value of the given expression is 2\\boxed{2}2\u200b.<\/p>\n\n\n\n

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

    We used known trigonometric identities to simplify the terms. First, we used the identity cos\u206170\u2218=sin\u206120\u2218\\cos 70^\\circ = \\sin 20^\\circcos70\u2218=sin20\u2218 to cancel out terms in the first fraction. In the second part, the identity sin\u206154\u2218=cos\u206136\u2218\\sin 54^\\circ = \\cos 36^\\circsin54\u2218=cos36\u2218 allowed us to simplify the expression for cos\u206136\u2218csc\u206154\u2218\\cos 36^\\circ \\csc 54^\\circcos36\u2218csc54\u2218, leading to a value of 1. Finally, the sum of the two parts results in 2.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    \\frac{\\cos 70^\\circ}{\\sin 20^\\circ} + \\cos 36^\\circ \\csc 54^\\circ The Correct Answer and Explanation is: The expression we are looking to simplify is:cos\u206170\u2218sin\u206120\u2218+cos\u206136\u2218csc\u206154\u2218\\frac{\\cos 70^\\circ}{\\sin 20^\\circ} + \\cos 36^\\circ \\csc 54^\\circsin20\u2218cos70\u2218\u200b+cos36\u2218csc54\u2218 Step-by-Step Simplification: Final Answer: So, the entire expression simplifies as follows:1+1=21 + 1 = 21+1=2 Thus, the value of the given expression is 2\\boxed{2}2\u200b. Explanation: We […]<\/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-264783","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/264783","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=264783"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/264783\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=264783"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=264783"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=264783"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}