If $\sin 3 \theta = \sin \theta$,how many solutions exist such that $-2 \pi < \theta < 2 \pi$?

Solve the following equation $\sin x + \sqrt{3} \cos x = \sqrt{2}$.

If the sum of all the solutions of the equation $8 \cos x \cdot \left( \cos \left( \frac{\pi}{6} + x \right) \cdot \cos \left( \frac{\pi}{6} - x \right) - \frac{1}{2} \right) = 1$ in the interval $[0, \pi]$ is $k\pi$,then $k$ is equal to:

The sum of all $x \in [0, \pi]$ which satisfy the equation $\sin x + \frac{1}{2} \cos x = \sin^2(x + \frac{\pi}{4})$ is

If $\cos(\alpha - \beta) = 1$ and $\cos(\alpha + \beta) = \frac{1}{e}$,where $-\pi < \alpha, \beta < \pi$,then the total number of ordered pairs $(\alpha, \beta)$ is:

The general solution of $\tan \theta + \tan 2\theta = \tan 3\theta$ is