If $\tan ^{-1} x + \tan ^{-1} y + \tan ^{-1} z = \frac{\pi}{2}$,where $x, y, z > 0$ and $xy < 1$,then the value of $xy + yz + zx$ is:

The value of $\cot ^{-1}\left(\frac{\sqrt{1+\tan ^2(2)}-1}{\tan (2)}\right)-\cot ^{-1}\left(\frac{\sqrt{1+\tan ^2\left(\frac{1}{2}\right)}+1}{\tan \left(\frac{1}{2}\right)}\right)$ is equal to

If $\tan^{-1} \frac{1}{1+1(2)} + \tan^{-1} \frac{1}{1+2(3)} + \tan^{-1} \frac{1}{1+3(4)} + \dots + \tan^{-1} \frac{1}{1+n(n+1)} = \tan^{-1} \theta$,then $\theta$ =

If $0 \leqslant \cos ^{-1} x \leqslant \pi$ and $\frac{-\pi}{2} \leqslant \sin ^{-1} x \leqslant \frac{\pi}{2}$,then at $x=\frac{1}{5}$ the value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ is

The trigonometric equation $\sin ^{-1} x = 2 \sin ^{-1} 2a$ has a real solution, if

Simplify $\tan ^{-1}\left[\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right],$ if $\frac{a}{b} \tan x > -1$.