Half Angle Formulas/Hyperbolic Tangent

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Theorem

Let $x \in \R$.

Then:

\(\ds \tanh \frac x 2\) \(=\) \(\ds +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) for $x \ge 1$
\(\ds \tanh \frac x 2\) \(=\) \(\ds -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) for $x \le 1$

where $\tanh$ denotes hyperbolic tangent and $\cosh$ denotes hyperbolic cosine.


Corollary 1

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


Corollary 2

For $x \ne 0$:

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


Proof

\(\ds \tanh \frac x 2\) \(=\) \(\ds \frac {\sinh \frac x 2} {\cosh \frac x 2}\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac {\pm \sqrt {\frac {\cosh x - 1} 2} } {\pm \sqrt {\frac {\cosh x + 1} 2} }\) Half Angle Formula for Hyperbolic Sine and Half Angle Formula for Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\)


We also have that:

when $x \ge 0$, $\tanh \dfrac x 2 \ge 0$
when $x \le 0$, $\tanh \dfrac x 2 \le 0$.

$\blacksquare$


Also see


Sources

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