Finding a composite function

Question

Given that $h(x) = f(g(x))$

$h(x) = \sin(x)$

$f(x) = \cos(x)$

What is the function $g(x)$?

Answer 1

Over a certain set, you could use $g(x)=\cos^{-1}(\sin(x)),$ since then $f(g(x)=\cos(\cos^{-1}(\sin(x)))=\sin(x),$ since the cos and cos inverse cancel.

Answer 2

$$h(x) = f(g(x))=\cos (g(x))=\sin x=cos(x-\frac{\pi}{2})$$ so $ g(x)=2k\pi+x-\frac{\pi}{2}$ or $g(x)=2k\pi-(x-\frac{\pi}{2})$