Half Angle Formula for Hyperbolic Tangent/Corollary 1

Theorem
Let $x \in \R$.

Then:
 * $\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$

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

Proof
Since $\cosh x > 0$, it follows that $\cosh x + 1 > 0$.

We also have that:
 * when $x \ge 0$, $\tanh \dfrac x 2 \ge 0$ and $\sinh x \ge 0$
 * when $x \le 0$, $\tanh \dfrac x 2 \le 0$ and $\sinh x \le 0$.

Thus:
 * $\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$

Also see

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