# Hyperbolic Cosine Function is Even

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## Theorem

Let $\cosh: \C \to \C$ be the hyperbolic cosine function on the set of complex numbers.

Then $\cosh$ is even:

$\map \cosh {-x} = \cosh x$

## Proof 1

 $\displaystyle \map \cosh {-x}$ $=$ $\displaystyle \frac {e^{-x} + e^{-\paren {-x} } } 2$ Definition of Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \frac {e^{-x} + e^x} 2$ $\displaystyle$ $=$ $\displaystyle \frac {e^x + e^{-x} } 2$ $\displaystyle$ $=$ $\displaystyle \cosh x$

$\blacksquare$

## Proof 2

 $\displaystyle \map \cosh {-x}$ $=$ $\displaystyle \map \cos {-i x}$ Hyperbolic Cosine in terms of Cosine $\displaystyle$ $=$ $\displaystyle \map \cos {i x}$ Cosine Function is Even $\displaystyle$ $=$ $\displaystyle \cosh x$ Hyperbolic Cosine in terms of Cosine

$\blacksquare$