Any ideas for tanh(x/2)

**Catholicman** · April 20th 2010, 07:27 AM

Any idéa how I can veryfy the tanh(x/2) formula?
It is tanh(x/2) = (coshx-1)/(sinhx) but I cant prove it..

**General** · April 20th 2010, 07:59 AM

The Right Hand Side $=\frac{cosh(x)-1}{sinh(x)}$

$=\frac{cosh(x)}{sinh(x)} - \frac{1}{sinh(x)}$

$=coth(x)-csch(x)$

$=\frac{e^x+e^{-x}}{e^x-e^{-x}} - \frac{2}{e^x-e^{-x}}$

$=\frac{e^x+e^{-x}-2}{e^x-e^{-x}} \, \cdot \, \frac{e^x}{e^x}$

$=\frac{e^{2x}-2e^x+1}{e^{2x}-1}$

$=\frac{(e^x-1)^2}{(e^x-1)(e^x+1)}$

$=\frac{e^x-1}{e^x+1} \, \cdot \, \frac{e^{-\frac{x}{2}}}{e^{-\frac{x}{2}}}$

$=\frac{ e^{\frac{x}{2}}-e^{-\frac{x}{2}} }{ e^{\frac{x}{2}}+e^{-\frac{x}{2}} }$

$=tanh(\frac{x}{2})$ $=$ The Left Hand Side.

**Soroban** · April 20th 2010, 08:32 AM

Hello, Catholicman!

We need a few hyperbolic identities:

. . $\cosh^2\!z - \sinh^2\!z \:=\:1 \quad\Rightarrow\quad \cosh^2\!z - 1 \:=\:\sinh^2\!z$

. . $\tanh z \:=\:\frac{\sinh z}{\cosh z}$

. . $\sinh\left(\frac{z}{2}\right) \:=\:\sqrt{\frac{\cosh z - 1}{2}}$

. . $\cosh\left(\frac{z}{2}\right) \;=\;\sqrt{\frac{\cosh z + 1}{2}}$

Verify: . $\tanh\left(\frac{x}{2}\right) \;=\; \frac{\cosh x-1}{\sinh x}$

We have: . $\tanh\left(\frac{x}{2}\right) \;=\;\frac{\sinh(\frac{x}{2})}{\cosh(\frac{x}{2})} \;=\;\frac{\sqrt{\dfrac{\cosh x - 1}{2}}}{\sqrt{\dfrac{\cosh x+1}{2}}} \;=\;\sqrt{\frac{\cosh x-1}{\cosh x+1}}$

Multiply by $\frac{\cosh x -1}{\cosh x-1}\!:\;\;\; \sqrt{\frac{\cosh x-1}{\cosh x+1}\cdot{\color{blue}\frac{\cosh x-1}{\cosh x-1}}} \;=\; \sqrt{\frac{(\cosh x-1)^2}{\cosh^2x-1}}$

. . . . . . . . . . . . . . . . $=\;\sqrt{\frac{(\cosh x - 1)^2}{\sinh^2x}} \;=\;\frac{\cosh x - 1}{\sinh x}$

Therefore: . $\tanh\left(\frac{x}{2}\right) \;=\;\frac{\cosh x - 1}{\sinh x}$

**Catholicman** · April 20th 2010, 09:22 AM

Thankyou.

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