One form of the Robertson-Walker metric is $$ds^2 = c^2dt^2 - a(t)^2[d\chi^2+ S_k(\chi)^2(d\theta^2 + \sin^2\theta ~d\phi^2)]\tag{1}$$ $$\\$$
Considering curvature, where k = 0 , +1, -1 for flat, positive and negative curvatures respectively, then: $$S_k(\chi) = \begin{cases}R\sin(\chi/R)~~~~~~(k=+1)\\\chi~~~~~~~~~~~~~~~~~~~~~~~(k=0)\\R\sinh(\chi/R)~~~~(k=-1)\end{cases}$$ $$\\$$
But another form of this metric is $$ds^2=c^2dt^2 - a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2(d\theta^2+\sin^2~d\phi^2)\right]\tag{2}$$ $$\\$$
How does one go from $(1)$ to $(2)$? (or vice versa) I get that this is due to a switch of choice of coordinates, from radial to co-moving radial, I think? But how can it be shown that they represent the same metric?