$\tan ^{-1}(\cot x)+\cot ^{-1}(\tan x) =$ . . . . . .

Prove $\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)=\frac{x}{2}$,where $x \in\left(0, \frac{\pi}{4}\right)$.

If $y = 3 \sin^{-1}x + \sin^{-1}(3x - 4x^3)$ for all $x \in [-1/2, 1/2]$,then

The differential coefficient of $\cos^{-1} \left( \sqrt{\frac{1+x}{2}} \right)$ with respect to $x$ is

For $\theta \in \left(0, \frac{\pi}{2}\right)$,$\operatorname{sech}^{-1}(\cos \theta)$ is equal to

$\lim _{x \rightarrow 0^{+}} \frac{x \sin ^{-1}\left(\frac{2 x}{1+x^2}\right)}{\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) \tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)}$ is equal to