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jenyasd209 [6]
3 years ago
8

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

Mathematics
1 answer:
Tanzania [10]3 years ago
4 0

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

\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)

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