prove trigonometry identity for $\sin\alpha+\sin2\alpha+\sin3\alpha+...+\sin N\alpha$, whithout using induction

Question

prove that:

$$\sin \alpha+\sin2\alpha+\sin3\alpha+...+\sin N\alpha = \frac{\sin\frac{N\alpha}{2}\sin\frac{(N+1)\alpha}{2}}{\sin\frac{\alpha}{2}}$$ without $\sum$