Find the maximum value of the expression $\lfloor \tan^{-1} x - \tan^{-1} y \rfloor - \lfloor \sin^{-1} u - \sin^{-1} v \rfloor$,where $\lfloor . \rfloor$ denotes the greatest integer function,and $x, y, u, v$ are independent real variables.

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

If $2 \tan^{-1} x = 3 \sin^{-1} x$ and $x \neq 0$, then $8x^2 + 1 =$

If $\alpha = 3 \sin^{-1}\left(\frac{6}{11}\right)$ and $\beta = 3 \cos^{-1}\left(\frac{4}{9}\right)$,where the inverse trigonometric functions take only the principal values,then the correct option$(s)$ is(are):
$(A) \cos \beta > 0$
$(B) \sin \beta < 0$
$(C) \cos(\alpha + \beta) > 0$
$(D) \cos \alpha < 0$

If $\cos^{-1} x - \cos^{-1} \frac{y}{2} = \alpha$,where $-1 \le x \le 1$,$-2 \le y \le 2$,and $x \le \frac{y}{2}$,then for all $x, y$,$4x^2 - 4xy \cos \alpha + y^2$ is equal to

If $\tan (\cos ^{ - 1}x) = \sin (\cot ^{ - 1}\frac{1}{2})$,then $x =$