Question

Prove whether the following are identities 2tanh 1/2x / 1−tanh^2 1/2 x = sinh x​

Answer 1

Recall that

`\cosh^2(x) - \sinh^2(x) = 1`

Dividing both sides by cosh²(x) gives

`1 - \tanh^2(x) = \mathrm{sech}^2(x)`

Also, recall the identity

`\sinh(2x) = 2\sinh(x)\cosh(x)`

Then

`(2\tanh\left(\frac x2\right))/(1 - \tanh^2\left(\frac x2\right)) = \frac{2\tanh\left(\frac x2\right)}{\mathrm{sech}^2\left(\frac x2\right)} \\\\ (2\tanh\left(\frac x2\right))/(1 - \tanh^2\left(\frac x2\right)) = 2\tanh\left(\frac x2\right)\cosh^2\left(\frac x2\right) \\\\ (2\tanh\left(\frac x2\right))/(1 - \tanh^2\left(\frac x2\right)) = 2\sinh\left(\frac x2\right)\cosh\left(\frac x2\right) \\\\(2\tanh\left(\frac x2\right))/(1 - \tanh^2\left(\frac x2\right)) = \sinh(x)`