The value of $\tan \left(\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)$ is

Assertion $(A): \operatorname{cosech}^{-1}(3) = \log \left(\frac{1+\sqrt{10}}{3}\right)$
Reason $(R): e^{\operatorname{cosech}^{-1} x}$ is a root of the quadratic equation $x p^2 - 2p - x = 0$
The correct option among the following is

Considering the principal values of the inverse trigonometric functions,$\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}{2} \sqrt{1-x^2}\right)$,for $-\frac{1}{2} < x < \frac{1}{\sqrt{2}}$,is equal to:

Show that $\sin ^{-1}(2 x \sqrt{1-x^{2}})=2 \cos ^{-1} x$ for $\frac{1}{\sqrt{2}} \leq x \leq 1$.

$\sum\limits_{m = 1}^n {{{\tan }^{ - 1}}} \left( {\frac{{2m}}{{{m^4} + {m^2} + 2}}} \right)$ is equal to

$\sin \left[ \frac{\pi }{2} - \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \right] = $