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

If $0 \leq x \leq \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

$\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$ is equal to

If $\sin A + \sin 2A = x$ and $\cos A + \cos 2A = y,$ then $({x^2} + {y^2})({x^2} + {y^2} - 3) = $

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4}$. If $\sin (\alpha+\beta) = \frac{1}{3}$ and $\cos (\alpha-\beta) = \frac{2}{3}$,then the greatest integer less than or equal to $\left(\frac{\sin \alpha}{\cos \beta} + \frac{\cos \beta}{\sin \alpha} + \frac{\cos \alpha}{\sin \beta} + \frac{\sin \beta}{\cos \alpha}\right)^2$ is:

$\sin 20^{\circ}(4+\sec 20^{\circ})=$