Q92 / 150 · MHT CET 2022 (10 Aug Shift 2) · Mathematics

If $\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)$, then the value of $x$ is equal to

  • Correct answer
    A
    $-\frac{1}{2}$
  • B
    $-\frac{1}{3}$
  • C
    $\frac{1}{2}$
  • D
    $\frac{1}{3}$

Solution

$\begin{aligned} & \sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right) \\ & \Rightarrow \operatorname{sinsin}^{-1}\left\{\frac{1}{\sqrt{(x+1)^2+1}}\right\}=\operatorname{coscos}^{-1}\left\{\frac{1}{\sqrt{x^2+1}}\right\} \\ & \Rightarrow \sqrt{x^2+1}=\sqrt{x^2+2 x+2} \\ & \Rightarrow x=\frac{-1}{2}\end{aligned}$

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

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