Werner Formulas/Hyperbolic Sine by Hyperbolic Sine/Proof 1

Theorem

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

$\ds $

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds \frac {2 \sinh x \sinh y} 2$

$\ds $

$=$

$\ds \sinh x \sinh y$

$\blacksquare$