If $y = \cos^{-1}(\sin x)$, where $\frac{\pi}{2} < x < \pi$, then $\frac{dy}{dx} = \dots$

Let $\mathop {Lim}\limits_{x \to 0} \sec^{-1} \left( \frac{x}{\sin x} \right) = l$ and $\mathop {Lim}\limits_{x \to 0} \sec^{-1} \left( \frac{x}{\tan x} \right) = m$,then

If $x + \frac{1}{x} = 2$,the principal value of $\sin^{-1} x$ is

The principal value of ${\sin ^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right)$ is

$\tan^{-1} \frac{x}{\pi} < \frac{\pi}{3}, x \in N$,then the maximum value of $x$ is :-

Find the value of $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$.