If $\sin A, \cos A$ and $\tan A$ are in $G.P.$,then $\cos^3 A + \cos^2 A$ is equal to

If $\sin 10^{\circ} \sin 50^{\circ} \sin 60^{\circ} \sin 70^{\circ} = m$ and $\tan 20^{\circ} \tan 40^{\circ} \tan 60^{\circ} \tan 80^{\circ} = n$,then find the value of $\frac{n}{m}$.

For any positive integer $n$,let $S_n: (0, \infty) \rightarrow R$ be defined by $S_n(x) = \sum_{k=1}^n \cot^{-1}\left(\frac{1+k(k+1)x^2}{x}\right)$,where for any $x \in R$,$\cot^{-1} x \in (0, \pi)$ and $\tan^{-1} x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following statements is (are) $TRUE$?
$(A)$ $S_{10}(x) = \frac{\pi}{2} - \tan^{-1}\left(\frac{1+11x^2}{10x}\right)$,for all $x > 0$
$(B)$ $\lim_{n \rightarrow \infty} \cot(S_n(x)) = x$,for all $x > 0$
$(C)$ The equation $S_3(x) = \frac{\pi}{4}$ has a root in $(0, \infty)$
$(D)$ $\tan(S_n(x)) \leq \frac{1}{2}$,for all $n \geq 1$ and $x > 0$

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

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:

If $\frac{\sin (A+B)}{\sin (A-B)}=\frac{\cos (C+D)}{\cos (C-D)}$,then $\tan A \cot B=$