{"id":243929,"date":"2025-07-04T18:04:39","date_gmt":"2025-07-04T18:04:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243929"},"modified":"2025-07-04T18:04:41","modified_gmt":"2025-07-04T18:04:41","slug":"use-algebra-and-the-fundamental-trigonometric-identities","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/use-algebra-and-the-fundamental-trigonometric-identities\/","title":{"rendered":"Use algebra and the fundamental trigonometric identities."},"content":{"rendered":"\n

Simplify. csc(x) sec(x) Use algebra and the fundamental trigonometric identities. Your answer should be a number or use a single trigonometric function. cot(y) sin(x) cos(D) tan(D) cot(D) sec(D)<\/p>\n\n\n\n

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

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

Correct Answer:<\/strong> cot(x)<\/p>\n\n\n\n

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

The problem asks to simplify the trigonometric expression csc(x) \/ sec(x) into a single trigonometric function. This can be achieved by using fundamental trigonometric identities, specifically the reciprocal and quotient identities.<\/p>\n\n\n\n

Step 1: Apply Reciprocal Identities<\/strong><\/p>\n\n\n\n

The first step is to express csc(x) and sec(x) in terms of the more basic functions, sin(x) and cos(x). We use the reciprocal identities:<\/p>\n\n\n\n

    \n
  • The cosecant function is the reciprocal of the sine function:\u00a0csc(x) = 1 \/ sin(x)<\/li>\n\n\n\n
  • The secant function is the reciprocal of the cosine function:\u00a0sec(x) = 1 \/ cos(x)<\/li>\n<\/ul>\n\n\n\n

    By substituting these identities into the original expression, we get:<\/p>\n\n\n\n

    csc(x) \/ sec(x) = (1 \/ sin(x)) \/ (1 \/ cos(x))<\/p>\n\n\n\n

    Step 2: Simplify the Complex Fraction<\/strong><\/p>\n\n\n\n

    The expression is now a complex fraction (a fraction where the numerator, denominator, or both contain fractions). To simplify this, we use the algebraic rule for dividing fractions: dividing by a fraction is equivalent to multiplying by its reciprocal.<\/p>\n\n\n\n

    The reciprocal of the denominator (1 \/ cos(x)) is (cos(x) \/ 1). So, we can rewrite the expression as a multiplication:<\/p>\n\n\n\n

    (1 \/ sin(x)) * (cos(x) \/ 1)<\/p>\n\n\n\n

    Step 3: Perform the Multiplication<\/strong><\/p>\n\n\n\n

    Now, we multiply the numerators together and the denominators together:<\/p>\n\n\n\n

    (1 * cos(x)) \/ (sin(x) * 1) = cos(x) \/ sin(x)<\/p>\n\n\n\n

    Step 4: Apply the Quotient Identity<\/strong><\/p>\n\n\n\n

    The final step is to recognize that the resulting expression, cos(x) \/ sin(x), is the definition of another fundamental trigonometric function. The quotient identity for the cotangent function states:<\/p>\n\n\n\n

      \n
    • cot(x) = cos(x) \/ sin(x)<\/li>\n<\/ul>\n\n\n\n

      Therefore, we can replace cos(x) \/ sin(x) with cot(x).<\/p>\n\n\n\n

      Conclusion<\/strong><\/p>\n\n\n\n

      By systematically applying the reciprocal and quotient identities, we have simplified the expression\u00a0csc(x) \/ sec(x)\u00a0to a single trigonometric function,\u00a0cot(x). The answer satisfies the condition of being a single function.<\/p>\n\n\n\n

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

      Simplify. csc(x) sec(x) Use algebra and the fundamental trigonometric identities. Your answer should be a number or use a single trigonometric function. cot(y) sin(x) cos(D) tan(D) cot(D) sec(D) The Correct Answer and Explanation is: Correct Answer: cot(x) Explanation: The problem asks to simplify the trigonometric expression csc(x) \/ sec(x) into a single trigonometric function. This can be achieved […]<\/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-243929","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243929","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=243929"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243929\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243929"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243929"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243929"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}