Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine

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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

\(\ds \) \(\) \(\ds \frac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2\)
\(\ds \) \(=\) \(\ds \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \cosh \paren {x - y} } 2\) Hyperbolic Sine of Sum
\(\ds \) \(=\) \(\ds \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \paren {\sinh x \cosh y - \cosh x \sinh y} } 2\) Hyperbolic Sine of Difference
\(\ds \) \(=\) \(\ds \frac {2 \sinh x \cosh y} 2\)
\(\ds \) \(=\) \(\ds \sinh x \cosh y\)

$\blacksquare$


Also presented as

This result can also be seen presented as:

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


Sources

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