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JEE Main 20139 AprilOnline
Exam
JEE Main
Year
2013
Date
9 April
Mode
Online
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Main 2013 · 9 April · Online · Mathematics · Trigonometric Equations

A value of x for which ( ^ -1 (1+x)) = ( ^ -1 x), is :

A value of xx for which sin⁡(cot⁡−1(1+x))\sin \left(\cot ^{-1}(1+x)\right)=\cos\left($\tan ^{-1} x\right), is :

Options

  1. A−12-\frac{1}{2}
  2. B1
  3. C0
  4. D12\frac{1}{2}

Correct answer

A. −12-\frac{1}{2}

Step-by-step solution

sin⁡(cot⁡−1(1+x))=cos⁡(tan⁡−1x)\sin \left(\cot ^{-1}(1+x)\right)=\cos \left(\tan ^{-1} x\right)⇒cosec⁡2(cot⁡−1(1+x))=sec⁡2(tan⁡−1x)⇒1+[cot⁡(cot⁡−1(1+x))]2⇒=1+[sec⁡(tan⁡−1x)]2⇒(1+x)2=x2⇒x=−12\begin{aligned} & \Rightarrow \operatorname{cosec}^2\left(\cot ^{-1}(1+x)\right)=\sec ^2\left(\tan ^{-1} x\right) \\ & \Rightarrow 1+\left[\cot \left(\cot ^{-1}(1+x)\right)\right]^2 \\ & \Rightarrow=1+\left[\sec \left(\tan ^{-1} x\right)\right]^2 \\ & \Rightarrow(1+x)^2=x^2 \Rightarrow x=-\frac{1}{2}\end{aligned}$$
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