$\sum\limits_{r = 1}^{89} {{\log _3}(\tan {r^o})} = $

$16 \sin 12^{\circ} \cos 18^{\circ} \sin 48^{\circ} = $

The expression $(1 + \tan x + \tan^2 x)(1 - \cot x + \cot^2 x)$ has positive values for $x$,given by

If $\operatorname{cosec} \theta = \frac{p+q}{p-q}$,then $\cot \left(\frac{\pi}{4} + \frac{\theta}{2}\right)$ is equal to

Let $|\cos \theta \cos (60^{\circ}-\theta) \cos (60^{\circ}+\theta)| \leq \frac{1}{8}$, where $\theta \in [0, 2\pi]$. Then, the sum of all $\theta \in [0, 2\pi]$ where $\cos 3\theta$ attains its maximum value is: (in $\pi$)

If $x \sin^3 \alpha + y \cos^3 \alpha = \sin \alpha \cos \alpha$ and $x \sin \alpha - y \cos \alpha = 0$,then $x^2 + y^2 = $