The value of $\sin ^{-1}\left(-\frac{1}{\sqrt{2}}\right)+\cos ^{-1}\left(-\frac{1}{2}\right)-\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\tan ^{-1}(\sqrt{3})$ is:

$\tan ^{-1}\left[\frac{1}{\sqrt{3}} \sin \frac{5 \pi}{2}\right] + \sin ^{-1}\left[\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)\right]$ is equal to

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$,then the value of $x^{2025}+x^{2026}+x^{2027}$ is

If $y = \sec^{-1}\left( \frac{x + 1}{x - 1} \right) + \sin^{-1}\left( \frac{x - 1}{x + 1} \right)$,then $\frac{dy}{dx} = $

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

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 $ab < 1$.