To expand on Pete's comment where we have additional structure, in any context where duality makes sense, to show that $A=B$, it suffices to show that they both have the same duals. Of course this does not answer the question, since we still need a way of showing that their duals are equal, but sometimes this may be easier. Examples of duals are when $A$ and $B$ are both subsets of a common universe $U$ and the dual is just the complement. Or, if $A$ and $B$ are both subspaces of $R^n$ and the dual is the orthogonal complement.

Edit: Although I'm not an expert, I'm guessing there are examples coming from topological structure as well. For example, to show that a subset $A$ of a topological space $X$ is equal to $X$, it suffices to show that $A$ is closed and contains a dense subset of $X$.

To expand on Pete's comment where we have additional structure, in any context where duality makes sense, to show that $A=B$, it suffices to show that they both have the same duals. Of course this does not answer the question, since we still need a way of showing that their duals are equal, but sometimes this may be easier. Examples of duals are when $A$ and $B$ are both subsets of a common universe $U$ and the dual is just the complement. Or, if $A$ and $B$ are both subspaces of $R^n$ and the dual is the orthogonal complement.