The number of solutions of the equation $(4-\sqrt{3}) \sin x - 2 \sqrt{3} \cos^2 x = -\frac{4}{1+\sqrt{3}}$ for $x \in [-2\pi, \frac{5\pi}{2}]$ is

The number of values of $x$ satisfying $2\sin^2(2x) = 2\cos^2(8x) + \cos(10x)$ in the interval $x \in \left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$ is:

The most general value of $\theta$ which satisfies both the equations $\sin \theta = -\frac{1}{2}$ and $\tan \theta = \frac{1}{\sqrt{3}}$ is:

The number of values of $x$ in the interval $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ for which $14 \operatorname{cosec}^{2} x - 2 \sin^{2} x = 21 - 4 \cos^{2} x$ holds,is

If $\sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=1$,then find the value of $x$ in the interval $[0, \pi]$.

If $\sec 4\theta - \sec 2\theta = 2$, then $\theta =$