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:
$\displaystyle \tanh A = \frac{e^A - e^{-A}}{e^A + e^{-A}}$
$\displaystyle sech A = \frac{2}{e^A + e^{-A}}$
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.
Hello, geton!
Are we allowed to use basic hyperbolic identities?
Prove: .$\displaystyle \text{tanh}^2 A + \text{sech}^2A \:=\:1$
We have: . $\displaystyle \frac{\sinh^2A}{\cosh^2A} + \frac{1}{\cosh^2A} \;\;=\;\;\frac{\sinh^2A + 1}{\cosh^2A} \;\;=\;\;\frac{\cosh^2A}{\cosh^2A} \;\;=\;\; 1$