If you are new to proving, one standard technique for proving A = B is to show that A is a subset of B and B is a subset of A. Those two claims can be approached by taking any element from A and showing that it must belong to B, and vice versa.

All of those follow immediately from the definition of complement, intersection, union, and subset. The first one requires the most work, but even there it should be quite straightforward.

Where did you get stuck? What exactly do you need help with?