Answer
See the proof below.
Work Step by Step
We have
$$y'=\sec y \Longrightarrow \frac{dy}{\sec y}=dx\Longrightarrow \cos y dy=dx.$$
Now, by integration we get
$$\sin y =x+c.$$
When $x=0$, $y=0$, and the constant $c=0$. Hence,
$$\sin y =x\Longrightarrow y=\sin^{-1}x.$$
This completes the proof.