Half Angle Formulas/Hyperbolic Tangent/Corollary 2

Theorem
For $x \ne 0$:


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

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

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

Also see

 * Half Angle Formula for Hyperbolic Sine
 * Half Angle Formula for Hyperbolic Cosine