Sum of Cosines of Twice 2 Angles minus Cosine of Twice Third Angle of Triangle

Theorem
Let $\triangle ABC$ be a triangle.

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

Proof
First we note that:

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

Then: