If $\tan (x + y) = 33$ and $x = \tan^{-1}(3)$,then $y$ will be

$\sec ^2(\tan ^{-1} 2)+\operatorname{cosec}^2(\cot ^{-1} 3) = $

The value of $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{8}\right)$ is

Prove $\tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)=\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x$,where $-\frac{1}{\sqrt{2}} \leq x \leq 1$.

The smallest and the largest values of ${\tan ^{ - 1}}\left( {\frac{{1 - x}}{{1 + x}}} \right)$ for $0 \le x \le 1$ are

If $2 \sin^{-1} x - 3 \cos^{-1} x = 4$, then $2 \sin^{-1} x + 3 \cos^{-1} x =$