If $A + B + C = 270^o,$ then $\cos 2A + \cos 2B + \cos 2C + 4\sin A \sin B \sin C = $

If $b \sin \alpha = a \sin (\alpha + 2\beta)$,then $\frac{a + b}{a - b} = $

If $\left| a\sin^2 \theta + b\sin \theta \cos \theta + c\cos^2 \theta - \frac{1}{2}(a + c) \right| \le \frac{1}{2}k,$ then $k^2$ is equal to

$\sin^{2} \theta = \frac{(x+y)^{2}}{4xy}$ is possible only when

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

$\frac{{\sin 3\theta - \cos 3\theta }}{{\sin \theta + \cos \theta }} + 1 = $