# Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine

## Theorem

$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$

where $\cosh$ denotes hyperbolic cosine.

## Proof

 $\displaystyle$  $\displaystyle 2 \map \cosh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}$ $\displaystyle$ $=$ $\displaystyle 2 \frac {\map \cosh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \cosh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2$ Simpson's Formula for Hyperbolic Cosine by Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \cosh \frac {2 x} 2 + \cosh \frac {2 y} 2$ $\displaystyle$ $=$ $\displaystyle \cosh x + \cosh y$

$\blacksquare$

## Linguistic Note

The word prosthaphaeresis or prosthapheiresis is a neologism coined some time in the $16$th century from the two Greek words:

Ian Stewart, in his Taming the Infinite from $2008$, accurately and somewhat diplomatically describes the word as "ungainly".