Half Angle Formula for Hyperbolic Tangent/Corollary 2
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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
\(\ds \tanh \frac x 2\) | \(=\) | \(\ds \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\) | Half Angle Formula for Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\paren {\cosh x + 1} \paren {\cosh x - 1} } }\) | multiplying numerator and denominator by $\sqrt {\cosh x - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\cosh^2 x - 1} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\sinh^2 x} }\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \pm \frac {\cosh x - 1} {\sinh x}\) |
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