{"id":238178,"date":"2025-06-17T21:11:29","date_gmt":"2025-06-17T21:11:29","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238178"},"modified":"2025-06-17T21:11:31","modified_gmt":"2025-06-17T21:11:31","slug":"if-the-following-exression-represents-an-identity-1-sin2x%c3%b7cos2x-a-then-value-of-a-is-given-by","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/if-the-following-exression-represents-an-identity-1-sin2x%c3%b7cos2x-a-then-value-of-a-is-given-by\/","title":{"rendered":"If the following exression represents an identity 1-sin2x\u00f7cos2x =A then value of A is given by"},"content":{"rendered":"\n
If the following exression represents an identity 1-sin2x\u00f7cos2x =A then value of A is given by<\/pre>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
We are given the trigonometric expression:1\u2212sin\u20612xcos\u20612x=A\\frac{1 – \\sin 2x}{\\cos 2x} = Acos2x1\u2212sin2x\u200b=A<\/p>\n\n\n\n
We are to find the value of AAA and explain the identity.<\/p>\n\n\n\n
\n\n\n\n
Step-by-step Solution:<\/h3>\n\n\n\n
We start by recalling a useful trigonometric identity:tan\u2061\u03b8=sin\u2061\u03b8cos\u2061\u03b8\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}tan\u03b8=cos\u03b8sin\u03b8\u200b<\/p>\n\n\n\n
But that does not directly help. Instead, consider separating the terms in the expression:A=1\u2212sin\u20612xcos\u20612xA = \\frac{1 – \\sin 2x}{\\cos 2x}A=cos2x1\u2212sin2x\u200b<\/p>\n\n\n\n
There is no direct simplification unless we recall that sine and cosine of double angles can be expressed using basic identities, but since there is no simplification of 1\u2212sin\u20612×1 – \\sin 2×1\u2212sin2x in a way that would cancel with cos\u20612x\\cos 2xcos2x, we try another approach.<\/p>\n\n\n\n
We can try breaking this into separate terms using basic algebra:A=1cos\u20612x\u2212sin\u20612xcos\u20612xA = \\frac{1}{\\cos 2x} – \\frac{\\sin 2x}{\\cos 2x}A=cos2x1\u200b\u2212cos2xsin2x\u200b<\/p>\n\n\n\n
This separates the original expression into two parts:A=sec\u20612x\u2212tan\u20612xA = \\sec 2x – \\tan 2xA=sec2x\u2212tan2x<\/p>\n\n\n\n
So, the simplified form of the expression is:sec\u20612x\u2212tan\u20612x\\boxed{\\sec 2x – \\tan 2x}sec2x\u2212tan2x\u200b<\/p>\n\n\n\n
\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
A=sec\u20612x\u2212tan\u20612xA = \\boxed{\\sec 2x – \\tan 2x}A=sec2x\u2212tan2x\u200b<\/p>\n\n\n\n
\n\n\n\n
Explanation:<\/h3>\n\n\n\n
The given expression is:1\u2212sin\u20612xcos\u20612x\\frac{1 – \\sin 2x}{\\cos 2x}cos2x1\u2212sin2x\u200b<\/p>\n\n\n\n
To simplify this expression and find what it is equal to, we use algebraic manipulation and trigonometric identities. The first step is to break the single fraction into two separate terms. We write:1\u2212sin\u20612xcos\u20612x=1cos\u20612x\u2212sin\u20612xcos\u20612x\\frac{1 – \\sin 2x}{\\cos 2x} = \\frac{1}{\\cos 2x} – \\frac{\\sin 2x}{\\cos 2x}cos2x1\u2212sin2x\u200b=cos2x1\u200b\u2212cos2xsin2x\u200b<\/p>\n\n\n\n
Now, look at each term separately. The first term, 1cos\u20612x\\frac{1}{\\cos 2x}cos2x1\u200b, is a standard trigonometric identity. It is equal to sec\u20612x\\sec 2xsec2x, which is the reciprocal of cosine. The second term, sin\u20612xcos\u20612x\\frac{\\sin 2x}{\\cos 2x}cos2xsin2x\u200b, is also a known identity. It simplifies to tan\u20612x\\tan 2xtan2x, since tangent is the ratio of sine to cosine.<\/p>\n\n\n\n
Combining both simplified expressions gives us:sec\u20612x\u2212tan\u20612x\\sec 2x – \\tan 2xsec2x\u2212tan2x<\/p>\n\n\n\n
This means the original expression is equivalent to this simplified trigonometric form. Therefore, the identity is:1\u2212sin\u20612xcos\u20612x=sec\u20612x\u2212tan\u20612x\\frac{1 – \\sin 2x}{\\cos 2x} = \\sec 2x – \\tan 2xcos2x1\u2212sin2x\u200b=sec2x\u2212tan2x<\/p>\n\n\n\n
Identities like these are important in trigonometry because they allow us to rewrite complex expressions in simpler or more familiar forms. These simplifications help solve equations, integrate functions, or analyze wave behavior. Understanding these identities also strengthens your skill in recognizing patterns and relationships between trigonometric functions.<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
If the following exression represents an identity 1-sin2x\u00f7cos2x =A then value of A is given by The Correct Answer and Explanation is: We are given the trigonometric expression:1\u2212sin\u20612xcos\u20612x=A\\frac{1 – \\sin 2x}{\\cos 2x} = Acos2x1\u2212sin2x\u200b=A We are to find the value of AAA and explain the identity. Step-by-step Solution: We start by recalling a useful trigonometric […]<\/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-238178","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238178","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=238178"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238178\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}