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

Question

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

Aicharya · Accepted Answer

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

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

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