The number of solutions of the equation $\left( 1 - \frac{1}{2 \sin x} \right) \cos^2 2x = 2 \sin x - 3 + \frac{1}{\sin x}$ in the interval $[0, 4\pi]$ is:

The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

If in a triangle $ABC$,$AB=5$ units,$\angle B=\cos ^{-1}\left(\frac{3}{5}\right)$ and the radius of the circumcircle of $\triangle ABC$ is $5$ units,then the area (in sq. units) of $\triangle ABC$ is:

In a triangle $ABC$,if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$,then $\cos A : \cos B : \cos C =$

The angles of a triangle are in $A.P.$ and the greatest angle is double the least angle. Find the sine of the third angle.

Suppose that the sides $a, b, c$ of a triangle $ABC$ satisfy $b^2 = ac$. Then the set of all possible values of $\frac{\sin A \cot C + \cos A}{\sin B \cot C + \cos B}$ is