Theorem: Trigonometric Identities for Function Products
Sine, cosine, tangent and cotangent have the following properties:
\begin{align*}\cos \theta \cos \varphi &= \frac{1}{2}(\cos(\theta - \varphi) + \cos(\theta + \varphi)) \\ \\ \sin \theta \sin \varphi &= \frac{1}{2}(\cos(\theta - \varphi) - \cos(\theta + \varphi)) \\ \\ \sin \theta \cos \varphi &= \frac{1}{2}(\sin(\theta + \varphi) + \sin(\theta - \varphi)) \\ \\ \cos \theta \sin \varphi &= \frac{1}{2}(\sin(\theta + \varphi) - \sin(\theta - \varphi)) \\ \\ \tan \theta \tan \varphi &= \frac{\cos(\theta - \varphi) - \cos (\theta + \varphi)}{\cos(\theta - \varphi) + \cos (\theta + \varphi)} \\ \\ \tan \theta \cot \varphi &= \frac{\sin(\theta + \varphi) + \sin(\theta - \varphi)}{\sin(\theta + \varphi) - \sin(\theta - \varphi)}\end{align*}
PROOF
TODO