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Math Help - Hyperbolic Identity problem

  1. #1
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    Hyperbolic Identity problem

    Wasn't too sure where to put this one, so feel free to yell at me if it's in the wrong forum.

    I'm having some difficulty starting this problem,

    Show that

    (\cosh x + \sinh x)^k + (\cosh x - \sinh x)^k = 2\cosh kx

    I thought about proof by induction, or expanding the left hand side using the binomial theorem. But both times I got stuck. Any help would be much appreciated. Thanks

    Stonehambey
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Stonehambey View Post
    Wasn't too sure where to put this one, so feel free to yell at me if it's in the wrong forum.

    I'm having some difficulty starting this problem,

    Show that

    (\cosh x + \sinh x)^k + (\cosh x - \sinh x)^k = 2\cosh kx

    I thought about proof by induction, or expanding the left hand side using the binomial theorem. But both times I got stuck. Any help would be much appreciated. Thanks

    Stonehambey
    Here are some big hints :
    \cosh(x)=\frac{e^x+e^{-x}}{2}
    \sinh(x)=\frac{e^x-e^{-x}}{2}
    Thus \cosh(x)+\sinh(x)=\dots and \cosh(x)-\sinh(x)=\dots


    And remember that (x^a)^b=x^{ab}
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  3. #3
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    You only have to put in equation this definitions:

    cosh(x)=\frac{e^x+e^{-x}}{2}

    sinh(x)=\frac{e^x-e^{-x}}{2}


    (cosh(x)+sinh(x))^k+(cosh(x)-sinh(x))^k=e^{kx}+e^{-kx}=2cosh(kx)

    That's all,

    Have a nice day!
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  4. #4
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    *facepalms*

    Talk about not being able to see the wood for the trees!

    Thanks guys
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