Use the complex definition of $\sin z$ to find an expression for $\sin^{-1} z$
Solution 1:
As you defined $u = \sin z$, so $z = \arcsin u$. So,
$\arcsin u = \frac{1}{i}\ln\left(iu + \sqrt{1-u^2}\right)$.
Note $u$ is a dummy variable that can be renamed to other name without change the expression. So, $\arcsin z = \frac{1}{i}\ln\left(iz + \sqrt{1-z^2}\right)$.