Hyperbolic Sine Function is Odd/Proof 1

Theorem

$\map \sinh {-x} = -\sinh x$


Proof

\(\ds \map \sinh {-x}\) \(=\) \(\ds \frac {e^{-x} - e^{-\paren {-x} } } 2\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac {e^{-x} - e^x} 2\)
\(\ds \) \(=\) \(\ds -\frac {e^x - e^{-x} } 2\)
\(\ds \) \(=\) \(\ds -\sinh x\)

$\blacksquare$

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