A strange trigonometric identity in a proof of Niven's theorem I can't understand the inductive step on Lemma A in this proof of Niven's theorem. It asserts, where $n$ is an integer:
$$2\cos ((n-1)t)\cos (t) = \cos (nt) + \cos ((n-2)t)$$
I tried applying the angle subtraction formula to both sides, but all that does is introduce a bunch of sines, which I can't see how to eliminate.

As 
$\cos(A-B)+\cos(A+B)=\cos A\cos B+\sin A\sin B+\cos A\cos B-\sin A\sin B=2\cos A\cos B$
Put $A+B=nt,A-B=(n-2)t$
Alternatively use $\cos C+\cos D=2\cos\frac{C+D}2\cos\frac{C-D}2$
