Sum of Cosines of Twice Angles of Triangle

Theorem
Let $\triangle ABC$ be a triangle.

Then:
 * $\cos 2 A + \cos 2 B + \cos 2 C = -1 - 4 \cos A \cos B \cos C$

Proof
First we note that:

That is, $C$ is the supplement of $A + B$.

Then:

Also presented as
This result can also be presented as:


 * $\cos 2 A + \cos 2 B + \cos 2 C + 1 = -4 \cos A \cos B \cos C$