I'm trying to prove this with the following identities:
$\cos(a)+\cos(b)=2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2})$
$\cos(a)-\cos(b)=-2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2})$
Whenever I try to reduce the term like $2\cos((n-1)x)\cos(x)$ first, I try to set $x = \frac{a-b}{2}$ and use the first equality, but I don't know how to use the expression $\cos((n-1)\frac{a-b}{2})\cos(\frac{a-b}{2})$.