If $\sin \alpha - \cos \alpha = m$ and $\sin 2 \alpha = n - m^2$,where $-\sqrt{2} \leq m \leq \sqrt{2}$,then $n$ is equal to

Let $a$ and $b$ be non-negative real numbers. If $\sin x + a \cos x = b$,then $|a \sin x - \cos x| = $

Evaluate: $\sin 6^{\circ} + \sin 54^{\circ} + \sin 126^{\circ} + \cos 156^{\circ}$

The value of $\cos^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{4} + \cos^2 \frac{5\pi}{12}$ is

Let $x=a \sin ^\alpha \theta \cos ^{\alpha+1} \theta$ and $y=a \sin ^{\alpha+1} \theta \cos ^\alpha \theta$,where $\theta \neq \frac{n \pi}{2}$. If $\frac{(x^2+y^2)^m}{(xy)^n}$ is independent of $\theta$,then the relation between $\alpha, m$ and $n$ is:

Evaluate: $\cos^2 76^\circ + \cos^2 16^\circ - \cos 76^\circ \cos 16^\circ$