Werner Formulas/Hyperbolic Sine by Hyperbolic Sine/Proof 3

Theorem

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

$\ds \sinh x \sinh y$

$=$

$\ds i^2 \map \sin {\frac x i} \map \sin {\frac y i}$

$\ds $

$=$

$\ds -\map \sin {\frac x i} \map \sin {\frac y i}$

$i^2 = -1$

$\ds $

$=$

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

$\ds $

$=$

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

simplifying

$\ds $

$=$

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

$\blacksquare$