{"id":243918,"date":"2025-07-04T18:00:30","date_gmt":"2025-07-04T18:00:30","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243918"},"modified":"2025-07-04T18:00:32","modified_gmt":"2025-07-04T18:00:32","slug":"find-the-equivalent-expression","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/find-the-equivalent-expression\/","title":{"rendered":"Find the equivalent expression"},"content":{"rendered":"\n<p>Find the equivalent expression. csc x csc x cot x Csc x(csc * cot x) sec x(sec * tan x) CSC_ CSC * cot x) d. sec x(sec * tan x) sec x(sec * cot.)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To clarify your question, you&#8217;re asking to find the equivalent expression for a given trigonometric expression involving cosecant (csc), cotangent (cot), and secant (sec) functions. Here&#8217;s how we can break it down and simplify the expression:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Expression:<\/h3>\n\n\n\n<p>csc\u2061(x)\u22c5csc\u2061(x)\u22c5cot\u2061(x)\\csc(x) \\cdot \\csc(x) \\cdot \\cot(x)csc(x)\u22c5csc(x)\u22c5cot(x)<\/p>\n\n\n\n<p>This simplifies as: csc\u20612(x)\u22c5cot\u2061(x)\\csc^2(x) \\cdot \\cot(x)csc2(x)\u22c5cot(x)<\/p>\n\n\n\n<p>Now, let&#8217;s rewrite it in a more manageable form for simplification and identify any possible trigonometric identities.<\/p>\n\n\n\n<p>We can use some standard trigonometric identities here:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>csc\u2061(x)=1sin\u2061(x)\\csc(x) = \\frac{1}{\\sin(x)}csc(x)=sin(x)1\u200b<\/li>\n\n\n\n<li>cot\u2061(x)=cos\u2061(x)sin\u2061(x)\\cot(x) = \\frac{\\cos(x)}{\\sin(x)}cot(x)=sin(x)cos(x)\u200b<\/li>\n<\/ul>\n\n\n\n<p>Thus, the expression becomes: csc\u20612(x)\u22c5cot\u2061(x)=(1sin\u20612(x))\u22c5(cos\u2061(x)sin\u2061(x))\\csc^2(x) \\cdot \\cot(x) = \\left( \\frac{1}{\\sin^2(x)} \\right) \\cdot \\left( \\frac{\\cos(x)}{\\sin(x)} \\right)csc2(x)\u22c5cot(x)=(sin2(x)1\u200b)\u22c5(sin(x)cos(x)\u200b)<\/p>\n\n\n\n<p>Simplifying further: =cos\u2061(x)sin\u20613(x)= \\frac{\\cos(x)}{\\sin^3(x)}=sin3(x)cos(x)\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Next Expression: sec\u2061(x)\u22c5sec\u2061(x)\u22c5tan\u2061(x)\\sec(x) \\cdot \\sec(x) \\cdot \\tan(x)sec(x)\u22c5sec(x)\u22c5tan(x)<\/h3>\n\n\n\n<p>Now, for the second part of your question: sec\u20612(x)\u22c5tan\u2061(x)\\sec^2(x) \\cdot \\tan(x)sec2(x)\u22c5tan(x)<\/p>\n\n\n\n<p>Using the identity:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sec\u2061(x)=1cos\u2061(x)\\sec(x) = \\frac{1}{\\cos(x)}sec(x)=cos(x)1\u200b<\/li>\n\n\n\n<li>tan\u2061(x)=sin\u2061(x)cos\u2061(x)\\tan(x) = \\frac{\\sin(x)}{\\cos(x)}tan(x)=cos(x)sin(x)\u200b<\/li>\n<\/ul>\n\n\n\n<p>This expression becomes: sec\u20612(x)\u22c5tan\u2061(x)=(1cos\u20612(x))\u22c5(sin\u2061(x)cos\u2061(x))\\sec^2(x) \\cdot \\tan(x) = \\left( \\frac{1}{\\cos^2(x)} \\right) \\cdot \\left( \\frac{\\sin(x)}{\\cos(x)} \\right)sec2(x)\u22c5tan(x)=(cos2(x)1\u200b)\u22c5(cos(x)sin(x)\u200b)<\/p>\n\n\n\n<p>Simplifying: =sin\u2061(x)cos\u20613(x)= \\frac{\\sin(x)}{\\cos^3(x)}=cos3(x)sin(x)\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Equivalent Expressions:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>For csc\u20612(x)\u22c5cot\u2061(x)\\csc^2(x) \\cdot \\cot(x)csc2(x)\u22c5cot(x), the equivalent expression is: cos\u2061(x)sin\u20613(x)\\frac{\\cos(x)}{\\sin^3(x)}sin3(x)cos(x)\u200b<\/li>\n\n\n\n<li>For sec\u20612(x)\u22c5tan\u2061(x)\\sec^2(x) \\cdot \\tan(x)sec2(x)\u22c5tan(x), the equivalent expression is: sin\u2061(x)cos\u20613(x)\\frac{\\sin(x)}{\\cos^3(x)}cos3(x)sin(x)\u200b<\/li>\n<\/ol>\n\n\n\n<p>These two expressions are now simplified forms of the original ones you provided.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-373.jpeg\" alt=\"\" class=\"wp-image-243926\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the equivalent expression. csc x csc x cot x Csc x(csc * cot x) sec x(sec * tan x) CSC_ CSC * cot x) d. sec x(sec * tan x) sec x(sec * cot.) The Correct Answer and Explanation is: To clarify your question, you&#8217;re asking to find the equivalent expression for a given [&hellip;]<\/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-243918","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243918","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=243918"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243918\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243918"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243918"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243918"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}