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Math Help - | sinh (z) |^2 = sinh^2(x) + sin^2(y)

  1. #1
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    | sinh (z) |^2 = sinh^2(x) + sin^2(y)

    I am having trouble proving this identity |sinh(z)|^2=sinh^2(x)+sin^2(y) \ \ z\in\mathbb{C} \ \ x,y\in\mathbb{R}.

    \displaystyle \left(\sqrt{sinh^2(z)}\right)^2=\left(\frac{e^z-e^{-z}}{2}\right)^2=\frac{e^{2z}+e^{-2z}-2}{4}

    \displaystyle =\frac{e^{2x}cos(2y)+e^{-2x}cos(2y)-2+\mathbf{i}(e^{2x}sin(2y)-e^{-2x}sin(2y))}{4}

    \displaystyle =\frac{cos(2y)(e^{2x}+e^{-2x})-2+\mathbf{i}sin(2y)(e^{2x}-e^{-2x})}{4}

    \displaystyle =\frac{cos(2y)cosh(2x)-2+\mathbf{i}sin(2y)sinh(2x)}{2}

    At this point, I am not sure how to proceed.
    Last edited by dwsmith; December 21st 2010 at 01:29 PM. Reason: Changed progress to proceed.
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  2. #2
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    \sinh(z)=\sinh(x)\cos(y)+i~\cosh(x)\sin(y).
    |\sinh(z)|^2=\sinh^2(x)\cos^2(y)+\cosh^2(x)\sin^2(  y)
    What can you do with that?
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  3. #3
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    sinh^2(x)cos^2(y)+cosh^2(x)sin^2(y)

    cosh^2(x)=1+sinh^2(x), \ \ sinh^2(x)cos^2(y)+(1+sinh^2(x))sin^2(y)

    =sinh^2(x)(cos^2(y)+sin^2(y))+sin^2(y)=sinh^2(x)+s  in^2(y)
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