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