# Prosthaphaeresis Formulas/Hyperbolic Cosine minus Hyperbolic Cosine/Proof 1

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