Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
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Theorem
- $\coth x = \dfrac 1 {\tanh x}$
where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively.
Proof
\(\ds \coth x\) | \(=\) | \(\ds \frac {\cosh x} {\sinh x}\) | Definition 2 of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sinh x / \cosh x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\tanh x}\) | Definition 2 of Hyperbolic Tangent |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.8$: Relationships among Hyperbolic Functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions