Theorem: Trigonometric Identities for Function Sums
Sine, cosine, tangent and cotangent have the following properties:
\begin{align*}\sin \theta \pm \sin \varphi &= 2\sin \frac{\theta \pm \varphi}{2} \cos \frac{\theta \mp \varphi}{2} \\ \\ \cos \theta + \cos \varphi &= 2 \cos \frac{\theta + \varphi}{2} \cos \frac{\theta - \varphi}{2} \\ \\ \cos \theta - \cos \varphi &= -2 \sin \frac{\theta + \varphi}{2} \sin \frac{\theta - \varphi}{2} \\ \\ \tan \theta \pm \tan \varphi &= \frac{\sin(\theta \pm \varphi)}{\cos \theta \cos \varphi} \\ \\ \cot \theta \pm \cot \varphi &= \frac{\sin(\varphi \pm \theta)}{\sin \theta \sin \varphi}\end{align*}
PROOF
TODO