Trigonometry - Trigonometric Equations

Prove the identity \(\frac{2 \sin x + \sin^2 x}{1 - \cos 2x} = \operatorname{cosec} x + \frac{1}{2}\).

Hence solve the equation \(\frac{2 \sin x + \sin^2 x}{1 - \cos 2x} \tan^2 x = 0\), for \(-180^\circ \le x \le 180^\circ\).