a set of vertices $X \subseteq V(G)$ of a graph $G$ is independent if and only if its complement $V(G) \setminus X$ is a vertex cover. ( $V(G) \setminus X$ is complement of $X$ )
(Because if $X$ is VC, then for every edge atleast one endpoint will be in $X$. Removing $X$ from $V$ we get all nonadjacent vertices.)
Complement of vertex cover is Independent set and vice versa.
If that vertex cover is minimum, then complement is Maximum Independent set.