The simplest value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$ is

If $\sin \theta + \csc \theta = 2$,then $\sin^2 \theta + \csc^2 \theta = $

$\frac{\sin 70^\circ + \cos 40^\circ}{\cos 70^\circ + \sin 40^\circ} = $

$A$ 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 $ft$)

If $A = \sin^2 x + \cos^4 x$,then for all real $x :$

If $\cos 2B = \frac{\cos (A + C)}{\cos (A - C)}$,then $\tan A, \tan B, \tan C$ are in