The given expression is:cos\u2061(68\u2218)cos\u2061(38\u2218)+sin\u2061(68\u2218)sin\u2061(38\u2218)\\cos(68^\\circ)\\cos(38^\\circ) + \\sin(68^\\circ)\\sin(38^\\circ)cos(68\u2218)cos(38\u2218)+sin(68\u2218)sin(38\u2218)<\/p>\n\n\n\n
This is a trigonometric identity. It resembles the sum-to-product identity for cosine:cos\u2061(A)cos\u2061(B)+sin\u2061(A)sin\u2061(B)=cos\u2061(A\u2212B)\\cos(A)\\cos(B) + \\sin(A)\\sin(B) = \\cos(A – B)cos(A)cos(B)+sin(A)sin(B)=cos(A\u2212B)<\/p>\n\n\n\n
Here, A=68\u2218A = 68^\\circA=68\u2218 and B=38\u2218B = 38^\\circB=38\u2218. Applying the identity:cos\u2061(68\u2218)cos\u2061(38\u2218)+sin\u2061(68\u2218)sin\u2061(38\u2218)=cos\u2061(68\u2218\u221238\u2218)\\cos(68^\\circ)\\cos(38^\\circ) + \\sin(68^\\circ)\\sin(38^\\circ) = \\cos(68^\\circ – 38^\\circ)cos(68\u2218)cos(38\u2218)+sin(68\u2218)sin(38\u2218)=cos(68\u2218\u221238\u2218)<\/p>\n\n\n\n
Now, simplify the angle:68\u2218\u221238\u2218=30\u221868^\\circ – 38^\\circ = 30^\\circ68\u2218\u221238\u2218=30\u2218<\/p>\n\n\n\n
So, the expression simplifies to:cos\u2061(30\u2218)\\cos(30^\\circ)cos(30\u2218)<\/p>\n\n\n\n
The exact value of cos\u2061(30\u2218)\\cos(30^\\circ)cos(30\u2218) is known to be:cos\u2061(30\u2218)=32\\cos(30^\\circ) = \\frac{\\sqrt{3}}{2}cos(30\u2218)=23\u200b\u200b<\/p>\n\n\n\n
Therefore, the exact value of the original expression is:32\\frac{\\sqrt{3}}{2}23\u200b\u200b<\/p>\n\n\n\n
Explanation:<\/h3>\n\n\n\n
This solution leverages a fundamental trigonometric identity to simplify the given expression. The sum-to-product identity helps convert the sum of cosines and sines into a simpler cosine expression with a difference of angles. Recognizing that cos\u2061(30\u2218)=32\\cos(30^\\circ) = \\frac{\\sqrt{3}}{2}cos(30\u2218)=23\u200b\u200b allows for a straightforward evaluation. This identity is particularly useful when dealing with sums of trigonometric functions with angles that are easily recognizable.<\/p>\n\n\n\n Find the exact value of cos 68 cos 38 + sin 68 sin 38. The Correct Answer and Explanation is: The given expression is:cos\u2061(68\u2218)cos\u2061(38\u2218)+sin\u2061(68\u2218)sin\u2061(38\u2218)\\cos(68^\\circ)\\cos(38^\\circ) + \\sin(68^\\circ)\\sin(38^\\circ)cos(68\u2218)cos(38\u2218)+sin(68\u2218)sin(38\u2218) This is a trigonometric identity. It resembles the sum-to-product identity for cosine:cos\u2061(A)cos\u2061(B)+sin\u2061(A)sin\u2061(B)=cos\u2061(A\u2212B)\\cos(A)\\cos(B) + \\sin(A)\\sin(B) = \\cos(A – B)cos(A)cos(B)+sin(A)sin(B)=cos(A\u2212B) Here, A=68\u2218A = 68^\\circA=68\u2218 and B=38\u2218B = 38^\\circB=38\u2218. Applying the identity:cos\u2061(68\u2218)cos\u2061(38\u2218)+sin\u2061(68\u2218)sin\u2061(38\u2218)=cos\u2061(68\u2218\u221238\u2218)\\cos(68^\\circ)\\cos(38^\\circ) […]<\/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-244926","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244926","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=244926"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244926\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244926"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244926"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244926"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-549.jpeg\")
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