I'm watching this video lecture http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/ and I'm stuck at around 3:40, I can't seem to figure out what he is doing.
He is showing how to derive $f(x)=\sin^{-1}(x)$.
At some point he goes from the expression $\frac{dy}{dx}=\frac{1}{\cos(y)}$ to $\frac{dy}{dx}= \frac{1}{\sqrt{1-x^2}}$.
Ok, I know that's the result, that's how I always did it, but I never actually derived it myself. So yeah, I'd like to know what he did in those last two steps, to get from the first expression to the second.
I know it may (will) be something completely stupid and I'll say 'oh... facepalm', but for some reason I can't figure out how he did it. I'm guessing the next logical step is to replace $y=\sin^{-1}(x)$, but then? I've never been in good terms with trigonometric functions and identities really, so I'd appreciate some enlightening.
Thank you.