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

If $\cos A+\cos (A+B)+\cos (A+2 B)+\ldots$ up to $n$ terms $=$ $\cos \left(\frac{2 A+(n-1) B}{2}\right) \sin \frac{n B}{2} \operatorname{cosec} \frac{B}{2}$,then $\cos \frac{\pi}{19}+\cos \frac{3 \pi}{19}+\cos \frac{5 \pi}{19}+\ldots+\cos \frac{17 \pi}{19} = $

If $\cos x+\cos y=p$ and $\sin x+\sin y=q$,then $\cos \left(\frac{x-y}{2}\right) = $

Number of real values of $x \in (0, \pi)$ for which $\frac{8}{3\sin x - \sin 3x} + 3\sin^2 x \le 5$ is:

If $\sin \theta = \frac{1}{2} (x + \frac{1}{x})$,then $\sin 3 \theta + \frac{1}{2} (x^3 + \frac{1}{x^3}) = $

$\cos 12^{\circ} + \cos 60^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ} = $