Werner Formulas/Hyperbolic Sine by Hyperbolic Sine

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Theorem

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

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


Proof 1

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

$\blacksquare$


Proof 2

\(\ds \sinh x \sinh y\) \(=\) \(\ds \frac {e^x - e^{-x} } 2 \frac {e^y - e^{-y} } 2\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac {e^{x + y} - e^{x - y} - e^{-x + y} + e^{-x - y} } 4\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\dfrac {e^{x + y} + e^{-\paren {x + y} } } 2 - \frac {e^{x - y} + e^{-\paren {x - y} } } 2}\) rearranging
\(\ds \) \(=\) \(\ds \frac {\cosh \paren {x + y} - \cosh \paren {x - y} } 2\) Definition of Hyperbolic Cosine

$\blacksquare$


Proof 3

\(\ds \sinh x \sinh y\) \(=\) \(\ds i^2 \map \sin {\frac x i} \map \sin {\frac y i}\) Sine in terms of Hyperbolic Sine
\(\ds \) \(=\) \(\ds -\map \sin {\frac x i} \map \sin {\frac y i}\) $i^2 = -1$
\(\ds \) \(=\) \(\ds -\frac {\map \cos {\frac x i - \frac y i} - \map \cos {\frac x i + \frac y i} } 2\) Werner Formula for Sine by Sine
\(\ds \) \(=\) \(\ds \frac {\map \cos {\frac x i + \frac y i} - \map \cos {\frac x i - \frac y i} } 2\) simplifying
\(\ds \) \(=\) \(\ds \frac {\map \cosh {x + y} - \map \cosh {x - y} } 2\) Cosine in terms of Hyperbolic Cosine

$\blacksquare$


Also presented as

This result can also be seen presented as:

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


Sources

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