# Hyperbolic Cosine of Sum

## Theorem

$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$

where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine.

### Corollary

$\map \cosh {a - b} = \cosh a \cosh b - \sinh a \sinh b$

## Proof

 $\displaystyle$  $\displaystyle \cosh a \cosh b + \sinh a \sinh b$ $\displaystyle$ $=$ $\displaystyle \frac {e^a + e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a - e^{-a} } 2 \frac {e^b - e^{-b} } 2$ Definition of Hyperbolic Sine and Definition of Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \frac {e^{a + b} + e^{-a + b} + e^{a + b} + e^{-a - b} } 4$ Exponential of Sum $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac {e^{a + b} - e^{-a + b} - e^{a - b} + e^{-a - b} } 4$ $\displaystyle$ $=$ $\displaystyle \frac {2 e^{a + b} + 2 e^{-\paren {a + b} } } 4$ $\displaystyle$ $=$ $\displaystyle \frac {e^{a + b} + e^{-\paren {a + b} } } 2$ $\displaystyle$ $=$ $\displaystyle \map \cosh {a + b}$ Definition of Hyperbolic Cosine

$\blacksquare$