If $(\tan ^{-1} x)^2+(\cot ^{-1} x)^2=\frac{5 \pi^2}{8}$,then $x^2+1=$

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

$\tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{3} = ?$

Consider the following statements:
Assertion $(A)$: When $x, y, z$ are positive numbers, then $\operatorname{Tan}^{-1}\left(\sqrt{\frac{x(x+y+z)}{y z}}\right)+\operatorname{Tan}^{-1}\left(\sqrt{\frac{y(x+y+z)}{x z}}\right)+\operatorname{Tan}^{-1}\left(\sqrt{\frac{z(x+y+z)}{x y}}\right) = \pi$
Reason $(R)$: $\operatorname{Tan}^{-1} a + \operatorname{Tan}^{-1} b = \operatorname{Tan}^{-1}\left(\frac{a+b}{1-ab}\right)$ if $a > 0$ and $b > 0$ and $ab < 1$.

$2 \tan^{-1}(\cos x) = \tan^{-1}(\csc^2 x)$,then $x = $

The value of $\tan \left[2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right]$ is