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