$\frac{d}{dx}[\tan^{-1}(\cot x) + \cot^{-1}(\tan x)] = $

The value of $\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)$ for $|x| < \frac{1}{\sqrt{2}}, x \neq 0$ is:

If $y = \sin^{-1}\left(\frac{19}{20}x\right) + \cos^{-1}\left(\frac{19}{20}x\right)$,then $\frac{dy}{dx} = $

Prove $\cos ^{-1} \frac{4}{5} + \cos ^{-1} \frac{12}{13} = \cos ^{-1} \frac{33}{65}$

If $\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}$, then $1-x y-y z-z x$ is equal to

Let $a_1=1, a_2, a_3, a_4, \ldots$ be consecutive natural numbers. Then $\tan ^{-1}\left(\frac{1}{1+ a _1 a _2}\right)+\tan ^{-1}\left(\frac{1}{1+ a _2 a _3}\right)+\ldots+\tan ^{-1}\left(\frac{1}{1+ a _{2021} a _{2022}}\right)$ is equal to