# Prosthaphaeresis Formulas/Hyperbolic Cosine minus Hyperbolic Cosine

## Theorem

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

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

## Proof 1

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

$\blacksquare$

## Proof 2

 $\displaystyle 2 \, \map \sinh {\dfrac {x + y} 2} \, \map \sinh {\dfrac {x - y} 2}$ $=$ $\displaystyle \dfrac 1 2 \paren {e^{\frac {x + y} 2} - e^{-\frac {x + y} 2} } \paren {e^{\frac {x - y} 2} - e^{-\frac {x - y} 2} }$ Definition of Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \dfrac 1 2 \paren {e^x - e^y - e^{-y} + e^{-x} }$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {e^x + e^{-x} } 2 - \dfrac {e^y + e^{-y} } 2$ rearranging $\displaystyle$ $=$ $\displaystyle \cosh x - \cosh y$ Definition of Hyperbolic Cosine

$\blacksquare$

## 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".