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.

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