Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent

Theorem

$\coth x = \dfrac 1 {\tanh x}$

where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively.

$\ds \coth x$

$=$

$\ds \frac {\cosh x} {\sinh x}$

Definition 2 of Hyperbolic Cotangent

$\ds $

$=$

$\ds \frac 1 {\sinh x / \cosh x}$

$\ds $

$=$

$\ds \frac 1 {\tanh x}$

Definition 2 of Hyperbolic Tangent

$\blacksquare$

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