Question

Theorem: For any $x\in R$ ${sinh}^{-1}x={log}_e(x+\sqrt{x^2+1})$

Answer 1

Let $x=\sin hy$ …………$\left(1\right)$

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

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

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

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

$\Rightarrow e^{2y}-2x{e}^y-1=0$

Using quadratic formula,

$e^y=\frac{2x\pm \sqrt{(2x{)}^2+4}}{2}$

$=\frac{2x\pm 2\sqrt{x^2+1}}{2}$

$=x\pm \sqrt{x^2+1}$

Since $e^y$ is positive, $e^y=x+\sqrt{x^2+1}$

Taking logarithm to the base e on both sides.

$y=lo{g}_e(x+\sqrt{x^2+1})$

$\Rightarrow \sin h^{-1}x=lo{g}_e(x+\sqrt{x^2+1})$.