The value of $\sin ^{2} 75^{\circ}-\sin ^{2} 15^{\circ}$ is:

If $2y \cos \theta = x \sin \theta$ and $2x \sec \theta - y \csc \theta = 3$,then $x^2 + 4y^2 = $

If $0^{\circ} < \theta < 90^{\circ},$ the value of $\sin \theta + \cos \theta$ is

If $\cos ^{2} \theta-\sin ^{2} \theta=\frac{1}{3},$ where $0 \leq \theta \leq \frac{\pi}{2},$ then the value of $\cos ^{4} \theta-\sin ^{4} \theta$ 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 $m$)

$\sin ^2 \frac{\pi }{8} + \sin ^2 \frac{3\pi }{8} + \sin ^2 \frac{5\pi }{8} + \sin ^2 \frac{7\pi }{8} = $