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 derivative of $\tan^{-1} \left( \frac{x}{\sqrt{1 - x^2}} \right)$ with respect to $\sin^{-1}(x)$ is

If $\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}$,then $\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right)$ is equal to

Let the maximum value of $(sin^{-1}x)^{2} + (cos^{-1}x)^{2}$ for $x \in [-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}]$ be $\frac{m}{n}\pi^{2}$,where $\gcd(m, n) = 1$. Then $m+n$ is equal to ........... .

If $\sin \left( \sin^{-1} \frac{1}{5} + \cos^{-1} x \right) = 1$,then $x$ is equal to

If ${x^2} + {y^2} + {z^2} = {r^2}$,then ${\tan ^{ - 1}}\left( {\frac{{xy}}{{zr}}} \right) + {\tan ^{ - 1}}\left( {\frac{{yz}}{{xr}}} \right) + {\tan ^{ - 1}}\left( {\frac{{zx}}{{yr}}} \right) = $