Cosine of Sum/Proof 3

Theorem

 * $\cos \left({a + b}\right) = \cos a \cos b - \sin a \sin b$


 * $\sin \left({a + b}\right) = \sin a \cos b + \cos a \sin b$

where $\sin$ and $\cos$ are sine and cosine.

Proof
Recall the Sine Exponential Formulation:


 * $ \displaystyle \sin x = \frac 1 2 i \left( e^{-ix} - e^{ix} \right) $

Recall the Cosine Exponential Formulation:


 * $ \displaystyle \cos x = \frac 1 2 \left( e^{-ix} + e^{ix} \right) $

Then, starting from the RHS:

Similarly: