Q64 / 150 · MHT CET 2024 (04 May Shift 1) · Mathematics

If $\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)$ then considering positive square roots, $x$ has the value $\qquad$

  • A
    0
  • B
    $\frac{9}{4}$
  • C
    $\frac{1}{2}$
  • Correct answer
    D
    $-\frac{1}{2}$

Solution

$\begin{aligned} & \sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right) \\ & \sin \left[\sin ^{-1} \frac{1}{\sqrt{x^2+2 x+2}}\right]=\cos \left[\cos ^{-1} \frac{1}{\sqrt{1+x^2}}\right] \\ & \frac{1}{\sqrt{x^2+2 x+2}}=\frac{1}{\sqrt{1+x^2}} \\ & \Rightarrow \sqrt{1+x^2}=\sqrt{x^2+2 x+2} \\ & \Rightarrow 1+x^2=x^2+2 x+2 \\ & \Rightarrow 2 x+1=0 \\ & \Rightarrow x=\frac{-1}{2}\end{aligned}$

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

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