As an answer to the question Proof that for $a>0$ and $a + 1/a$ element of $\mathbb{Z}$, $a^n + 1/a^n$ is always element of $\mathbb{Z}$ by induction, user236182 gave this answer:
$$a^{k+1}+\frac{1}{a^{k+1}}=\left(a^{k}+\frac{1}{a^k}\right)\left(a+\frac{1}{a}\right)-\left(a^{k-1}+\frac{1}{a^{k-1}}\right) $$
This identity made me think of this one:
$\cos((n+1)x) = 2\cos(nx)\cos(x)-\cos(n-1)x$.
Show that these two identities are equivalent.
(This is a result neither deep nor profound.)