# Simpson's Formulas/Hyperbolic Sine by Hyperbolic Cosine

## Theorem

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

where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.

## Proof

 $\displaystyle$  $\displaystyle \frac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$ $\displaystyle$ $=$ $\displaystyle \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \cosh \paren {x - y} } 2$ Hyperbolic Sine of Sum $\displaystyle$ $=$ $\displaystyle \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \paren {\sinh x \cosh y - \cosh x \sinh y} } 2$ Hyperbolic Sine of Difference $\displaystyle$ $=$ $\displaystyle \frac {2 \sinh x \cosh y} 2$ $\displaystyle$ $=$ $\displaystyle \sinh x \cosh y$

$\blacksquare$

## Also presented as

This result can also be seen presented as:

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