trigonometric equation (number of solutions)

[man111]

(1) No. of solution of the equation $1+\sin x\cdot\sin^2\frac{x}{2} = 0$ ,Where $x\in \left[-\pi,\pi\right]$

[Руст]

because $|\sin y|\le 1$ we get $\sin x=-1, \sin \frac x2=\pm 1$ or $x=\frac{\pi}{2}+2k\pi$, but $\sin\frac x2 =\pm \frac{\sqrt 2}{2}.$ - contradition.

[man111]

Means No solution.

thanks.