Prove the given identity and, where appropriate, check the identity independently by Osborn’s rule.

tanh^2 A + sech^2 A ≡ 1

By which way I can solve this?

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- Mar 10th 2008, 03:20 AMgetonHyperbolic function
Prove the given identity and, where appropriate, check the identity independently by Osborn’s rule.

**tanh^2 A + sech^2 A ≡ 1**

By which way I can solve this? - Mar 10th 2008, 03:39 AMmr fantastic
Substitute the definitions of tanh A and sech A:

$\displaystyle \tanh A = \frac{e^A - e^{-A}}{e^A + e^{-A}}$

$\displaystyle sech A = \frac{2}{e^A + e^{-A}}$

Now start simplifying.

As for the check, are you allowed to assume the identity $\displaystyle \cosh^2 A - \sinh^2 A = 1$? I'd say the answer is YES since you're allowed to use Osborne's Rule. Then the check follows easily. - Mar 10th 2008, 09:27 AMSoroban
Hello, geton!

Are we allowed to use__basic__hyperbolic identities?

Quote:

Prove: .$\displaystyle \text{tanh}^2 A + \text{sech}^2A \:=\:1$

We have: . $\displaystyle \frac{\sinh^2A}{\cosh^2A} + \frac{1}{\cosh^2A} \;\;=\;\;\frac{\sinh^2A + 1}{\cosh^2A} \;\;=\;\;\frac{\cosh^2A}{\cosh^2A} \;\;=\;\; 1$