If $P = \tan 15^{\circ} + \cot 15^{\circ}$,$Q = \tan 22 \frac{1}{2}^{\circ} + \cot 22 \frac{1}{2}^{\circ}$ and $R = \sin 54^{\circ} + \sin 18^{\circ}$,then their ascending order is

The value of $(1-\cos \theta)(1+\cos \theta)(1+\cot^2 \theta)$,when $\theta = \frac{\pi}{15}$ is

Evaluate: $\sin 21^{\circ} \cos 9^{\circ}-\cos 84^{\circ} \cos 6^{\circ}$

$3\left[ \sin^4\left( \frac{3\pi}{2} - \alpha \right) + \sin^4(3\pi + \alpha) \right] - 2\left[ \sin^6\left( \frac{\pi}{2} + \alpha \right) + \sin^6(5\pi - \alpha) \right] = $

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4}$. If $\sin (\alpha+\beta) = \frac{1}{3}$ and $\cos (\alpha-\beta) = \frac{2}{3}$,then the greatest integer less than or equal to $\left(\frac{\sin \alpha}{\cos \beta} + \frac{\cos \beta}{\sin \alpha} + \frac{\cos \alpha}{\sin \beta} + \frac{\sin \beta}{\cos \alpha}\right)^2$ is:

If $\sin x + \sin y = p$ and $\cos x + \cos y = q$,then $\sec(x + y) = $