If $dx/dy =\sin(x)$ then is $dy/dx = 1/\sin(x)$?
Solution 1:
By the chain rule (assuming all quantities exist and make sense):
$$\dfrac {\mathrm dx}{\mathrm dy}\dfrac {\mathrm dy}{\mathrm dx} = \dfrac {\mathrm dx}{\mathrm dx} = 1$$
do you see how that answers your question?
edit: the OP asked when this works. This works if $y$ is invertible and the derivative isn't $0$. If $y$ is not invertible, then $x$ might still implicitly define a differentiable function of $y$ for some neighborhood of a point you're interested in.