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SSC CPO 2022 PYQ — Question 106

SSC CPO - 9 Nov 2022 - Shift 1 • Quantitative Aptitude • Q106

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If {(sec⁡θ−1sec⁡θ+1)}n=cosec θ−cot⁡θ,\left\{{\left(\frac {\sec \theta - 1}{\sec \theta + 1}\right)}\right\}^n= \textrm {cosec}\space \theta - \cot \theta, then n = ?

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Correct answer: B

Explanation

The correct answer is option 2 i.e. 0.5

Given that

{(sec⁡θ−1sec⁡θ+1)}n=cosec θ−cot⁡θ\left\{{\left(\frac {\sec \theta - 1}{\sec \theta + 1}\right)}\right\}^n= \textrm {cosec}\space \theta - \cot \theta

⇒ [(sec⁡θ−1)(sec⁡θ+1)(sec⁡θ−1)(sec⁡θ−1)]n=cosec θ−cot⁡θ[\frac {(\sec \theta - 1)}{(\sec \theta + 1)}\frac {(\sec \theta - 1)}{(\sec \theta - 1)}]^n= \textrm {cosec}\space \theta - \cot \theta

⇒ [(sec⁡θ−1)2(sec⁡2θ−1)]n=cosec θ−cot⁡θ[\frac {(\sec \theta - 1)^2}{(\sec^2 \theta - 1)}]^n= \textrm {cosec}\space \theta - \cot \theta

⇒ [(sec2 θ + 1 - 2 sec θ)/tan2 θ]n = cosec θ - cot θ

⇒ [(cosec2 θ + cot2θ  - 2 cosec θ cot θ)]n =  cosec θ - cot θ

⇒  [cosec θ - cot θ]2n =  cosec θ - cot θ

Hence, 2n = 1 ⇒ n = 1/2 = 0.5