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Math Help - Hyperbolic functions

  1. #1
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    Hyperbolic functions

    From the definition of sinhx and coshx, in terms of exponentials show that,

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

    Can anyone help explain how to solve this? Cheers.
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  2. #2
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    Quote Originally Posted by Haris View Post
    From the definition of sinhx and coshx, in terms of exponentials show that,

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

    Can anyone help explain how to solve this? Cheers.
     sinh(x) = \frac{e^x-e^{-x}}{2}

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

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

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

    Hence you're trying to prove that:


      \frac{e^x-e^{-x}}{2} + \frac{e^y-e^{-y}}{2} = 2 \times \frac{e^{\frac{x+y}{2}}-e^{-\frac{x+y}{2}}}{2} \times \frac{e^{\frac{x-y}{2}}+e^{-\frac{x-y}{2}}}{2}

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

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

    Can you go from here?
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  3. #3
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    2 \times \frac{e^{\frac{x+y}{2}}-e^{-\frac{x+y}{2}}}{2} \times \frac{e^{\frac{x-y}{2}}+e^{-\frac{x-y}{2}}}{2}
    " alt="
    \frac{e^x-e^{-x}}{2} + \frac{e^y-e^{-y}}{2} = 2 \times \frac{e^{\frac{x+y}{2}}-e^{-\frac{x+y}{2}}}{2} \times \frac{e^{\frac{x-y}{2}}+e^{-\frac{x-y}{2}}}{2}
    " />

    How did the 2 (highlighted in red) cancel out?
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  4. #4
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    It cancels because of the multiplication of the denominators:
    2/(2*2)= 1/2

    Therefore, the two cancels.
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