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Math Help - Hyperbolic Trig Proof Help

  1. #1
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    Hyperbolic Trig Proof Help

    Im stuck on the following problem and would greatly appreciate any help.

    Using the exponential definitions of coshx and sinhx


    sinhx = (e^x - e^-x)/2 and coshx = (e^x + e^-x)/2<br />

    show that

    sinhx + sinhy = 2sinh((x+y) /2)cosh((x-y)/2)

    I know that it obviously involves use of the double angle formula but just cant get it work. Cheers
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  2. #2
    MHF Contributor red_dog's Avatar
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    2\sinh\frac{x+y}{2}\cosh\frac{x-y}{2}=\displaystyle 2\cdot\displaystyle\frac{e^{\frac{x+y}{2}}-e^{\frac{-x-y}{2}}}{2}\cdot\frac{e^{\frac{x-y}{2}}+e^{\frac{-x+y}{2}}}{2}=

    =\frac{1}{2}\left(e^{\frac{x+y}{2}+\frac{x-y}{2}}+e^{\frac{x+y}{2}+\frac{-x+y}{2}}-e^{\frac{-x-y}{2}+\frac{x-y}{2}}-e^{\frac{-x+y}{2}+\frac{-x+y}{2}}\right)=

    =\frac{1}{2}(e^x+e^y-e^{-y}-e^{-x})=\frac{e^x-e^{-x}}{2}+\frac{e^y-e^{-y}}{2}=\sinh x+\sinh y
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  3. #3
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    awesome thanks very much
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