Derivative of Arcsine Function/Proof

Proof
Let $y = \arcsin x$ where $-1 < x < 1$.

Then:

Then:

Now $\cos y \ge 0$ on the image of $\arcsin x$, that is:
 * $y \in \closedint {-\dfrac \pi 2} {\dfrac \pi 2}$

Thus it follows that we need to take the positive root of $\sqrt {1 - \sin^2 y}$.

So: