Evaluate: $\sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3)$

If $\cos ^{-1}\left(\frac{1}{2}\right)=\cot \left(\cos ^{-1} x\right)$, then the value of $x$ is

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

Consider the following statements:
Statement-$I$: $\operatorname{Cosh}^{-1} x = \operatorname{Tanh}^{-1} x$ has no solution.
Statement-$II$: $\operatorname{Cosh}^{-1} x = \operatorname{Coth}^{-1} x$ has only one solution.
The correct answer is:

If $f(x) = \frac{\sqrt{\operatorname{Cos}^{-1} x}}{\sqrt{2(1-x)}}$ for $x < 1$,then $\lim_{x \rightarrow 1^{-}} f(x) =$

Considering only the principal values of an inverse function,the set $A = \{x \geq 0 \mid \tan^{-1} x + \tan^{-1} 6x = \frac{\pi}{4}\}$