# Cosine Exponential Formulation/Real Domain/Proof 2

## Theorem

$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$

## Proof

Recall Euler's Formula:

$e^{i x} = \cos x + i \sin x$

Then, starting from the right hand side:

 $\displaystyle \frac {e^{i x} + e^{-i x} } 2$ $=$ $\displaystyle \frac {\cos x + i \sin x + \cos \paren {-x} + i \sin \paren {-x} } 2$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\cos x + \cos \paren {-x} } 2$ $\quad$ Sine Function is Odd $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {2 \cos x} 2$ $\quad$ Cosine Function is Even $\quad$ $\displaystyle$ $=$ $\displaystyle \cos x$ $\quad$ $\quad$

$\blacksquare$