Hyperbolic Sine Function is Odd/Proof 1
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Theorem
- $\map \sinh {-x} = -\sinh x$
Proof
| \(\ds \map \sinh {-x}\) | \(=\) | \(\ds \frac {e^{-x} - e^{-\paren {-x} } } 2\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{-x} - e^x} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac {e^x - e^{-x} } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\sinh x\) |
$\blacksquare$