calculus - How does $frac{1}{cos y}$ = $frac{1}{sqrt{1-x^2}}$

I was going through a textbook of mine and I noticed that in the proof of derivative of

$y = \sin^{-1}$,

i.e. proof that

$$\frac{dy}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1-x^2}}$$

there's a point where they say

$$\frac{1}{\cos y} = \frac{1}{\sqrt{1-x^2}}$$

and I'm not sure how this makes sense or works out. I've looked for proofs, tried implicit differentiation, and tried graphing it, but couldn't find anything. Can someone please explain?