trigonometry - Prove this formula for $cos{ntheta}-cos{nalpha}$

If $n$ be any positive integer, prove that $$\cos{n\theta}-\cos{n\alpha}=2^{n-1}[\cos \theta - \cos \alpha]\left[\cos \theta -\cos \left(\alpha + \frac {2\pi}n\right)\right]\cdots\left[\cos \theta -\cos \left(\alpha + (n-1)\frac {2\pi}n\right)\right]$$

I am struggling to establish this result. I have tried induction, but it doesn't help. Could I have a hint?