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?
$\cos nx=T_n(\cos x)$ where $T_n$ is a Chebyshev polynomial of the first kind. This is a polynomial of degree $n$ with leading coefficient $1/2^{n-1}$. Your identity amounts to $$T_n(u)-\cos ny=\frac1{2^{n-1}}\prod_{k=0}^{n-1}\left(u -\cos\left(y+\frac{2\pi k}{n}\right)\right).$$ To prove this, one needs to prove that the zeroes of the polynomial $T_n(u)-\cos ny$ are the $\cos(y+2\pi k/n)$,