Find the value of $\cos \tan^{-1} \sin \cot^{-1} (x)$ . Find the value of $\cos \tan^{-1} \sin \cot^{-1} (x)$ .
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Let $\cot^{-1} x=z$
$$x=\cot z$$
Then,
$$\sin \cot^{-1} (x)$$
$$=\sin z$$
$$=\dfrac {1}{\csc z}$$
$$=\dfrac {1}{\sqrt {1+\cot^2 z}}$$
$$=\dfrac {1}{\sqrt {1+x^2}}$$

Then we have $$\cos\left(\left(\tan^{-1}\dfrac1{\sqrt{1+x^2}}\right)\right)$$
If $\tan^{-1}\dfrac1{\sqrt{1+x^2}}=y,\dfrac\pi4\le y\le\dfrac\pi2\implies\tan y=\dfrac1{\sqrt{1+x^2}}$ and $\cos y\ge0$
$\cos\left(\tan^{-1}\dfrac1{\sqrt{1+x^2}}\right)=\cos y=+\dfrac1{\sqrt{1+\tan^2y}}=?$
