The value of $\cos^{-1}(\cos 12) - \sin^{-1}(\sin 14)$ is

If $f(x) = \cot^{-1} \left( \frac{3x - x^3}{1 - 3x^2} \right)$ and $g(x) = \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)$,then for $0 < a < \frac{1}{\sqrt{3}}$,the value of $\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}$ is

If $\sin ^{-1} x + \sin ^{-1}(1-x) = \cos ^{-1} x$, then $x \in$

Let $f : R \to R$,$f(x) = \max\{|\tan^{-1}x|, \cot^{-1}x\}$. Consider the following statements:
$I.$ The function is continuous and derivable $\forall x \in R$.
$II.$ The range of the function is $\left[ \frac{\pi}{4}, \pi \right]$.
$III.$ $f(x)$ is many-one into.
Identify the correct option.

If ${\tan ^{ - 1}}x + {\cos ^{ - 1}}\frac{y}{{\sqrt {1 + {y^2}} }} = {\sin ^{ - 1}}\frac{3}{{\sqrt {10} }}$ and both $x$ and $y$ are positive integers,then the possible values of $(x, y)$ are:

The number of real solutions of the equation $\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \pi/2$ is