Theorem: Trigonometric Identities for Angle Sums
Sine, cosine, tangent and cotangent have the following properties:
\begin{align*}\sin(\theta \pm \varphi) &= \sin\theta\cos\varphi \pm \cos\theta \sin \varphi \\ \\ \cos (\theta \pm \varphi) &= \cos \theta \cos \varphi \mp \sin \theta \varphi \\ \\ \tan (\theta \pm \varphi) &= \frac{\tan \theta \pm \tan \varphi}{1 \mp \tan \theta \tan \varphi} \\ \\ \cot(\theta \pm \varphi) &= \frac{\cot \theta \cot \varphi \mp 1}{\cot \varphi \pm \cot \theta}\end{align*}
PROOF
TODO