Sine of Twice Angle minus Sum of Sines of Twice Other Two Angles of Triangle

Theorem
Let $\triangle ABC$ be a triangle.

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

Proof
First we note that:

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

Then: