If $\sin x + \sin y = 3(\cos y - \cos x),$ then the value of $\frac{\sin 3x}{\sin 3y}$ is

If $A$ lies in the second quadrant and $B$ lies in the third quadrant and $\cos A = -\frac{\sqrt{3}}{2}$ and $\sin B = -\frac{3}{5}$,then find the value of $\frac{2 \tan B + \sqrt{3} \tan A}{\cot^2 A + \cos B}$.

If $2 \sin \left(\frac{\pi}{2}\right) = x^{2} + \frac{1}{x^{2}}$,then the value of $\left(x - \frac{1}{x}\right)$ is

The shadow of a tower standing on a level plane is found to be $60\, m$ longer when the angle of elevation of the sun is $30^{\circ}$ than when it is $45^{\circ}$. Find the height of the tower.

Let $f(x) = \sin^2 x + \cos^4 x + 2$ and $g(x) = \cos(\cos x) + \cos(\sin x)$. If the periods of $f(x)$ and $g(x)$ are $T_1$ and $T_2$ respectively,then:

Find the value of $\tan 4^{\circ} \tan 43^{\circ} \tan 47^{\circ} \tan 86^{\circ}$.