Number of solutions to the system of equations $sin \frac{x+y}{2}=0$ and $|x| + |y| = 1$

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

If $x = \frac{{n\pi }}{2}$ , satisfies the equation $sin\, \frac{x}{2}- cos \frac{x}{2} = 1$ $- sin\, x$ & the inequality $\left| {\frac{x}{2}\,\, - \,\,\frac{\pi }{2}} \right|\,\, \le \,\,\frac{{3\pi }}{4}$, then:

For $0<\theta<\frac{\pi}{2}$, the solution(s) of $\sum_{m=1}^6 \operatorname{cosec}\left(\theta+\frac{(m-1) \pi}{4}\right) \operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right)=4 \sqrt{2}$ is(are)

$(A)$ $\frac{\pi}{4}$ $(B)$ $\frac{\pi}{6}$ $(C)$ $\frac{\pi}{12}$ $(D)$ $\frac{5 \pi}{12}$

If $sin^2x + sinx \,cosx -6cos^2x = 0$ and $-\frac{\pi}{2} < x < 0$, then the value of $cos2x$, is

The general solution of $\sin x - 3\sin 2x + \sin 3x = $ $\cos x - 3\cos 2x + \cos 3x$ is