Cosine Exponential Formulation/Proof 1

Proof
Recall the definition of the cosine function:


 * $\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$

Recall the definition of the exponential as a power series:


 * $\displaystyle e^x = \sum_{n \mathop = 0}^\infty \frac {x^n}{n!} = 1 + x + \frac {x^2} 2 + \frac {x^3} 6 + \cdots$

Then, starting from the RHS: