Answer
$u = arctan(\frac{x}{\sqrt{1-x^2}})$
Work Step by Step
$u = arcsin~x$
By the definition of arcsin:
$x = sin~u$
$sin~u = x = \frac{x}{1} = \frac{opp}{hyp}$
We can find an expression for the adjacent side:
$adj = \sqrt{1^2-x^2} = \sqrt{1-x^2}$
The opposite side is $x$
The adjacent side is $\sqrt{1-x^2}$
The hypotenuse is 1
We can write an expression for $tan~u$:
$tan~u = \frac{opp}{adj}$
$tan~u = \frac{x}{\sqrt{1-x^2}}$
We can solve this equation for $u$:
$u = arctan(\frac{x}{\sqrt{1-x^2}})$
