Question

Express $\sec h^{-1}x$ in logarithmic form for $0<x<1$.

Answer 1

Let $sec{h}^{-1}x=y$

$\Rightarrow x=sechy$

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

$\Rightarrow x{e}^{2y}-2e^y+x=0$

$\Rightarrow e^y=\frac{1\pm \sqrt{1-x^2}}{x}$

$\Rightarrow y=ln\left(\frac{1\pm \sqrt{1-x^2}}{x}\right)$