If $f(x) = \tan^{-1}\left\{ \frac{\log(e/x^2)}{\log(ex^2)} \right\} + \tan^{-1}\left( \frac{3 + 2\log x}{1 - 6\log x} \right)$,then $\frac{d^n y}{dx^n}$ is $(n \ge 1)$.

$4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239}$ 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 ........... .

With reference to the principal values,if $\sin ^{-1} x + \sin ^{-1} y + \sin ^{-1} z = \frac{3 \pi}{2}$,then $x^{100} + y^{100} + z^{100} =$

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 $ab < 1$.

$\cot ^{ - 1}\left[ \frac{\sqrt {1 - \sin x} + \sqrt {1 + \sin x}}{\sqrt {1 - \sin x} - \sqrt {1 + \sin x}} \right] = $