Half Angle Formulas/Hyperbolic Tangent/Corollary 2

From ProofWiki
Jump to navigation Jump to search

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