The value of $\cos \left( an ^{-1}\left(\sin \left(\cot ^{-1} x ight) ight) ight)$ is

Question

The value of $\cos \left(	an ^{-1}\left(\sin \left(\cot ^{-1} xight)ight)ight)$ is

Aicharya · Accepted Answer

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

The value of $\cos \left(\tan ^{-1}\left(\sin \left(\cot ^{-1} x\right)\right)\right)$ is

Solution