The minimum value of $\cos 2\theta + \cos \theta$ for real values of $\theta$ 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 $\sin \theta + \csc \theta = 2$,then $\sin^2 \theta + \csc^2 \theta = $

If $a \cos \theta + b \sin \theta = p$ and $a \sin \theta - b \cos \theta = q$,then the relation between $a, b, p,$ and $q$ is:

If $x = a \cos^3 \theta$ and $y = b \sin^3 \theta$,then:

If $\tan A + \sin A = m$ and $\tan A - \sin A = n,$ then $\frac{(m^2 - n^2)^2}{mn} = $