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Math Help - Hyperbolic function

  1. #1
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    Hyperbolic function

    Prove the given identity and, where appropriate, check the identity independently by Osbornís rule.

    tanh^2 A + sech^2 A ≡ 1



    By which way I can solve this?
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  2. #2
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    Quote Originally Posted by geton View Post
    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.
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  3. #3
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    Hello, geton!

    Are we allowed to use basic hyperbolic identities?


    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

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