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

$\blacksquare$