$\cot ^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right)=$ . . . . . . where,$x>1$.

If $y = \cot^{-1} \left( \frac{1 + x}{1 - x} \right)$,then $\frac{dy}{dx} = $

Given $0 \leq x \leq \frac{1}{2}$,then the value of $\tan \left[\sin ^{-1}\left\{\frac{x}{\sqrt{2}}+\frac{\sqrt{1-x^{2}}}{\sqrt{2}}\right\}-\sin ^{-1} x\right]$ is:

If ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = 3\pi ,$ then $xy + yz + zx = $

If $y = \tan^{-1} \sqrt{\frac{1 + \cos x}{1 - \cos x}}$,then $\frac{dy}{dx}$ is

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