Q47 / 90 · JEE Main 2013 (09 Apr Online) · Mathematics

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

  • Correct answer
    A
    $-\frac{1}{2}$
  • B
    1
  • C
    0
  • D
    $\frac{1}{2}$

Solution

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

Hence, the correct answer is option: 'A'.

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