If $\sin A, \cos A$ and $\tan A$ are in $G.P.$,then $\cos^3 A + \cos^2 A$ is equal to

If $\cot x = \frac{5}{12}$ for some $x \in (\pi, \frac{3\pi}{2})$, then $\sin 7x(\cos \frac{13x}{2} + \sin \frac{13x}{2}) + \cos 7x(\cos \frac{13x}{2} - \sin \frac{13x}{2})$ 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?

Given that $\pi < \alpha < \frac{3\pi}{2}$,then the expression $\sqrt{4\sin^4 \alpha + \sin^2 2\alpha} + 4\cos^2 \left(\frac{\pi}{4} - \frac{\alpha}{2}\right)$ is equal to

Let $f(x) = \frac{\sin(x-a) + \sin(x+a)}{\cos(x-a) - \cos(x+a)}$,then

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