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Math Help - Can somebody prove - Hyperbolic functions

  1. #1
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    Can somebody prove - Hyperbolic functions

    Could somebody please prove these identities and explain a little bit ? I just can't figure them out.

    sinh^-1x = ln(x + sq.rt of(x^2 + 1))

    cosh^-1x = ln(x + sq.rt of (x^2 + 1))

    tanh^-1x = (1/2)ln ( (1+x)/(1 - x) )
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  2. #2
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    Quote Originally Posted by yakkow
    Could somebody please prove these identities and explain a little bit ? I just can't figure them out.

    sinh^-1x = ln(x + sq.rt of(x^2 + 1))
    I prove just one, you prove the rest.
    Since,
    y=\sinh x is a bijective function it has an inverse.
    Thus you interchange the x and y in,
    y=\frac{e^x-e^{-x}}{2}
    To get,
    x=\frac{e^y-e^{-y}}{2}
    Thus,
    2x=e^y-e^{-y}
    Thus,
    e^{2y}-1=2xe^{y}
    Thus,
    e^{2y}-2xe^y-1=0
    Allow, t=e^y thus,
    t^2-2xt-1=0
    Thus,
    t=\frac{2x\pm\sqrt{4x^2+4}}{2}
    Thus,
    t=\frac{2x\pm 2\sqrt{x^2+1}}{2}
    Which is,
    t=x\pm \sqrt{x^2+1}
    Since, t>0 we have,
    t=x+\sqrt{x^2+1}
    Thus,
    e^y=x+\sqrt{x^2+1}
    Thus,
    \sinh^{-1}x=\ln (x+\sqrt{x^2+1})
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