Prove the Identity $\frac{\sin ^{3} A-\cos ^{3} A}{\sin ^{2} A-\cos ^{2} A}=\frac{2+\sin 2 A}{2(\sin A+\cos A)}$

Prove the identity:
$\frac{\sin ^{3} A-\cos ^{3} A}{\sin ^{2} A-\cos ^{2} A}=\frac{2+\sin 2 A}{2(\sin A+\cos A)}$