The sum of the minimum and maximum values of the function $f(x) = \cos^{-1}x + 2\cot^{-1}x - 2x^3 - 4x$ is:

Let $f(x) = \tan^{-1} (\cot x - 2 \cot 2x)$. Then $\left[ \sum_{r = 1}^7 f(r) \right]$ is equal to (where $[.]$ represents the greatest integer function).

Given that the inverse trigonometric function assumes principal values only. Let $x, y$ be any two real numbers in $[-1, 1]$ such that $\cos ^{-1} x - \sin ^{-1} y = \alpha$,where $-\frac{\pi}{2} \leq \alpha \leq \pi$. Then,the minimum value of $x^2 + y^2 + 2xy \sin \alpha$ is

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

$\sec ^2(\tan ^{-1} 3) + \operatorname{cosec}^2(\cot ^{-1} 3) = $ . . . . . . .

If $\alpha, \beta$ are the solutions of the equation $\operatorname{Sin}^{-1} x - \operatorname{Cos}^{-1} x = \operatorname{Sin}^{-1}(3x - 2)$ and $\alpha > \beta$,then $3\alpha + 4\beta =$