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Math Help - Problem proving arcsech(x)

  1. #1
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    Problem proving arcsech(x)

    I have a problem where i have to prove arcsech(x) = log[(1+sqr(1-X^2))/x]
    does anyone know how to do this?
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  2. #2
    Super Member Random Variable's Avatar
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    let  y= \text{sech} x = \frac{1}{\text{cosh} x} = \frac{2}{e^{x}+e^{-x}} and solve for  x

     ye^{x}+ye^{-x} - 2 = 0

    now let  u = e^{x}

    then  yu + y\frac{1}{u} - 2 = 0

    or  yu^{2} - 2u+y = 0

    using the quadratic equation to solve for  u

    u = \frac{2 \pm \sqrt{4-4y^{2}}}{2y} = \frac{1 \pm \sqrt{1-y^{2}}}{y}

     u is greater than zero (since  e^{x} > 0 ), so ignore the negative root

    then  e^{x} = \frac{1 + \sqrt{1-y^{2}}}{y}

    and  x = \ln \Big(\frac{1 + \sqrt{1-y^{2}}}{y}\Big) for  0 < y \le 1
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