Recall that
[tex]\cosh^2(x) - \sinh^2(x) = 1[/tex]
Dividing both sides by cosh²(x) gives
[tex]1 - \tanh^2(x) = \mathrm{sech}^2(x)[/tex]
Also, recall the identity
[tex]\sinh(2x) = 2\sinh(x)\cosh(x)[/tex]
Then
[tex]\dfrac{2\tanh\left(\frac x2\right)}{1 - \tanh^2\left(\frac x2\right)} = \dfrac{2\tanh\left(\frac x2\right)}{\mathrm{sech}^2\left(\frac x2\right)} \\\\ \dfrac{2\tanh\left(\frac x2\right)}{1 - \tanh^2\left(\frac x2\right)} = 2\tanh\left(\dfrac x2\right)\cosh^2\left(\dfrac x2\right) \\\\ \dfrac{2\tanh\left(\frac x2\right)}{1 - \tanh^2\left(\frac x2\right)} = 2\sinh\left(\dfrac x2\right)\cosh\left(\dfrac x2\right) \\\\\dfrac{2\tanh\left(\frac x2\right)}{1 - \tanh^2\left(\frac x2\right)} = \sinh(x)[/tex]
