Prove arcsin x for its logarithm form

[ichabodgrant]

Homework Statement

Given sin x = (e^ix - e^-ix) / 2i,
I want to prove that

arcsin x = -i ln(ix + √1 - x²)

Homework Equations

I know about the Euler's formula and complex number. But I have never learned about complex logarithms.

The Attempt at a Solution

I try to use x = sin y. But it seems not working. How to show that relation?

 

[jedishrfu]

Why not try working it backward? Take the arcsin x and substitute it into the first equation for x.