Let $x = \sin(2 \tan^{-1} \alpha)$ and $y = \sin(\frac{1}{2} \tan^{-1} \frac{4}{3})$. If $S = \{\alpha \in R : y^2 = 1 - x\}$,then $\sum_{\alpha \in S} 16 \alpha^3$ is equal to $...........$

If $a_1, a_2, a_3, \dots, a_n$ is an $A.P.$ with common difference $d$,then $\tan \left[ \tan^{-1} \left( \frac{d}{1 + a_1 a_2} \right) + \tan^{-1} \left( \frac{d}{1 + a_2 a_3} \right) + \dots + \tan^{-1} \left( \frac{d}{1 + a_{n-1} a_n} \right) \right] = $

If $y = \cos \left(\frac{\pi}{3} + \cos^{-1} \frac{x}{2}\right)$,then $(x - y)^2 + 3y^2$ is equal to . . . . . . .

If $x = \sin \left( 2 \tan^{-1} 2 \right)$ and $y = \sin \left( \frac{1}{2} \tan^{-1} \frac{4}{3} \right)$,then -

If $\alpha$ and $\beta$ (where $\alpha > \beta$) are two zeroes of the equation $3\cos^{-1}\left(x^2 - 5x - \frac{11}{2}\right) = \pi$,then $(\alpha^2 + \beta^3)$ is -

If $x, y, z$ are in $A.P.$ and $\tan^{-1} x, \tan^{-1} y, \tan^{-1} z$ are also in another $A.P.$,then: