If $0 \le x < \frac{\pi}{2}$,then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

If $\cos(\alpha - \beta) = 1$ and $\cos(\alpha + \beta) = 1/e$,where $\alpha, \beta \in [-\pi, \pi]$,the number of pairs of $(\alpha, \beta)$ which satisfy both equations is:

The general solution of the equation $\tan x + \tan 2x - \tan 3x = 0$ is

In $(0, 2\pi)$,the number of solutions of $\tan \theta + \sec \theta = 2 \cos \theta$ is:

If $\alpha$ is a root of the equation $25 \cos^2 \theta + 5 \cos \theta - 12 = 0$,for $\frac{\pi}{2} < \alpha < \pi$,then $\sin 2\alpha =$

If $\tan \theta + \tan 2\theta + \sqrt{3} \tan \theta \tan 2\theta = \sqrt{3},$ then