# Cosine Function is Even/Proof 2

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## Theorem

$\map \cos {-z} = \cos z$

That is, the cosine function is even.

## Proof

 $\displaystyle \cos \paren {-z}$ $=$ $\displaystyle \frac {e^{i \paren {-z} } + e^{-i \paren {-z} } } 2$ Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac {e^{i z} + e^{-i z} } 2$ simplifying $\displaystyle$ $=$ $\displaystyle \cos z$ Cosine Exponential Formulation

$\blacksquare$