If $\sin \theta + \cos \theta = 1$,then $\sin \theta \cos \theta = $

If $\sin \beta$ is the geometric mean between $\sin \alpha$ and $\cos \alpha,$ then $\cos 2\beta$ is equal to

If $\sec \theta = x + \frac{1}{4x}$ where $0^{\circ} < \theta < 90^{\circ}$,then $\sec \theta + \tan \theta$ is equal to:

The value of $\tan 1^{\circ} \tan 2^{\circ} \ldots \tan 89^{\circ}$ 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 $x + \frac{1}{x} = 2\cos \theta ,$ then ${x^3} + \frac{1}{{{x^3}}} = $