Werner Formulas/Hyperbolic Sine by Hyperbolic Sine/Proof 2

Theorem

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

$\ds \sinh x \sinh y$

$=$

$\ds \frac {e^x - e^{-x} } 2 \frac {e^y - e^{-y} } 2$

$\ds $

$=$

$\ds \frac {e^{x + y} - e^{x - y} - e^{-x + y} + e^{-x - y} } 4$

simplifying

$\ds $

$=$

$\ds \frac 1 2 \paren {\dfrac {e^{x + y} + e^{-\paren {x + y} } } 2 - \frac {e^{x - y} + e^{-\paren {x - y} } } 2}$

rearranging

$\ds $

$=$

$\ds \frac {\cosh \paren {x + y} - \cosh \paren {x - y} } 2$

$\blacksquare$