Simpson's Formulas/Hyperbolic Cosine by Hyperbolic Cosine

Theorem

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

where $\cosh$ denotes hyperbolic cosine.

Proof

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

$\blacksquare$

Also presented as

This result can also be seen presented as:

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