$\tan 15^\circ = $

If $\tan A - \tan B = x$ and $\cot B - \cot A = y,$ then $\cot (A - B) = $

$A$ ladder is placed along a wall such that its upper end is touching the top of the wall. The foot of the ladder is $10 \, ft$ away from the wall and the ladder makes an angle of $60^{\circ}$ with the ground. When a man starts climbing on it,it slips and now the ladder makes an angle of $30^{\circ}$ with the ground. How much did the ladder slip down from the top of the wall? (in $ft$)

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.

$\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \dots + \cos 180^\circ = $

$\frac{{\cos {{10}^o} + \sin {{10}^o}}}{{\cos {{10}^o} - \sin {{10}^o}}} = $