# Math Help - Problem proving arcsech(x)

1. ## 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?

2. 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 $0 < y \le 1$