The value of $\frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A}$ is

If $1+\sqrt{1+a}=(1+\sqrt{1-a}) \cot \alpha$ and $0 < a < 1$,then $\sin 4 \alpha=$

If $\cos \alpha + \cos \beta = a$,$\sin \alpha + \sin \beta = b$ and $\alpha - \beta = 2 \theta$,then $\frac{\cos 3 \theta}{\cos \theta} = $

Let $\theta, \phi \in [0, 2\pi]$ be such that $2 \cos \theta(1-\sin \phi) = \sin^2 \theta \left(\tan \frac{\theta}{2} + \cot \frac{\theta}{2}\right) \cos \phi - 1$,$\tan (2\pi - \theta) > 0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}$. Then $\phi$ cannot satisfy

Let $S$ be the set of all $\alpha \in \mathbb{R}$ such that the equation $\cos 2x + \alpha \sin x = 2\alpha - 7$ has a solution. Then $S$ is equal to

Statement $(S1): \sin 55^{\circ} + \sin 53^{\circ} - \sin 19^{\circ} - \sin 17^{\circ} = \cos 2^{\circ}$.
Statement $(S2):$ The range of $\frac{1}{3 - \cos 2x}$ is $\left[\frac{1}{4}, \frac{1}{2}\right]$.
Which one of the following is correct?