The general solution of $\frac{\tan 2x - \tan x}{1 + \tan x \tan 2x} = 1$ is:

If the solution of the equation $\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1$,where $x \in \left(0, \frac{\pi}{2}\right)$,is $\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$,where $\alpha, \beta$ are integers,then $\alpha+\beta$ is equal to:

The number of values of $\alpha$ in $[0, 2\pi]$ for which $2\sin^3\alpha - 7\sin^2\alpha + 7\sin\alpha = 2$ is:

The equation $\sin x + \cos x = 2$ has

If $\sin 6 \theta + \sin 4 \theta + \sin 2 \theta = 0$, then the general value of $\theta$ is

The number of solutions of the equation $\sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0$ for $\theta \in [-3\pi, 2\pi]$ is: