# Prosthaphaeresis Formulas/Hyperbolic Cosine minus Hyperbolic Cosine/Proof 2

$\cosh x - \cosh y = 2 \, \map \sinh {\dfrac {x + y} 2} \, \map \sinh {\dfrac {x - y} 2}$
 $\displaystyle 2 \, \map \sinh {\dfrac {x + y} 2} \, \map \sinh {\dfrac {x - y} 2}$ $=$ $\displaystyle \dfrac 1 2 \paren {e^{\frac {x + y} 2} - e^{-\frac {x + y} 2} } \paren {e^{\frac {x - y} 2} - e^{-\frac {x - y} 2} }$ 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
$\blacksquare$