The value of $\cot 12^{\circ} \cot 102^{\circ} + \cot 102^{\circ} \cot 66^{\circ} + \cot 66^{\circ} \cot 12^{\circ}$ is

$\sqrt{3} \operatorname{cosec} 20^{\circ} - \sec 20^{\circ}$ is equal to

The exact value of $\cos \frac{2\pi}{28} \csc \frac{3\pi}{28} + \cos \frac{6\pi}{28} \csc \frac{9\pi}{28} + \cos \frac{18\pi}{28} \csc \frac{27\pi}{28}$ is equal to

Let $\alpha, \beta$ and $\gamma$ be such that $0 < \alpha < \beta < \gamma < 2 \pi$. For any $x \in \mathbb{R}$,if $\cos (x+\alpha)+\cos (x+\beta)+\cos (x+\gamma)=0$,then $\tan (\gamma-\alpha) = $

If $\cos \theta - 4 \sin \theta = 1$,then $\sin \theta + 4 \cos \theta$ is equal to

Let $S = \left\{ \theta \in \left( 0, \frac{\pi}{2} \right) : \sum_{m=1}^{9} \sec \left( \theta + (m-1) \frac{\pi}{6} \right) \sec \left( \theta + \frac{m \pi}{6} \right) = -\frac{8}{\sqrt{3}} \right\}$. Then: