# Half Angle Formulas/Hyperbolic Tangent/Corollary 2

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## Contents

## Theorem

- $\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$

where $\tanh$ denotes hyperbolic tangent, $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.

## Proof

\(\displaystyle \tanh \frac x 2\) | \(=\) | \(\displaystyle \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) | Half Angle Formula for Hyperbolic Tangent | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pm \sqrt {\frac {\left({\cosh x - 1}\right)^2} {\left({\cosh x + 1}\right) \left({\cosh x - 1}\right)} }\) | multiplying top and bottom by $\sqrt {\cosh x - 1}$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pm \sqrt {\frac {\left({\cosh x - 1}\right)^2} {\cosh^2 x - 1} }\) | Difference of Two Squares | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pm \sqrt {\frac {\left({\cosh x - 1}\right)^2} {\sinh^2 x} }\) | Difference of Squares of Hyperbolic Cosine and Sine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \pm \frac {\cosh x - 1} {\sinh x}\) |

Since $\cosh x \ge 1$, it follows that $\cosh x - 1 \ge 0$, with equality happening at $x = 0$.

We also have that:

- When $x > 0$, $\tanh \dfrac x 2 > 0$ and $\sinh x > 0$
- When $x < 0$, $\tanh \dfrac x 2 < 0$ and $\sinh x < 0$.

Thus:

- $\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$

$\blacksquare$

## Also see

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.29$: Double Angle Formulas