If $\tan (A + B) = p$ and $\tan (A - B) = q,$ then the value of $\tan 2A$ in terms of $p$ and $q$ is

The angle of elevation of the top of a vertical tower situated perpendicularly on a plane is observed as $60^{\circ}$ from a point $P$ on the same plane. From another point $Q$,$10\, m$ vertically above the point $P$,the angle of depression of the foot of the tower is $30^{\circ}$. The height of the tower is (in $m$)

If $\tan A + \cot A = 2$,then the value of $\tan^{10} A + \cot^{10} A$ is

$\tan 15^\circ = $

If $\cos (\alpha - \beta ) = 1$ and $\cos (\alpha + \beta ) = \frac{1}{e}$,where $-\pi < \alpha, \beta < \pi$,then the total number of ordered pairs $(\alpha, \beta)$ is:

If $\sin \alpha = \frac{8}{17}, 0 < \alpha < 90^{\circ}$ and $\tan \beta = \frac{5}{12}, 0 < \beta < 90^{\circ},$ then $\cos (\alpha - \beta)$ is