# Hyperbolic function

• Mar 10th 2008, 04:20 AM
geton
Hyperbolic 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, 04:39 AM
mr fantastic
Quote:

Originally Posted by geton
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?

Substitute the definitions of tanh A and sech A:

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

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

Now start simplifying.

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

Are we allowed to use basic hyperbolic identities?

Quote:

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

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