If the union and intersection of two NP languages are both in P, prove that the langauges are in co-NP
Hint: You only need to distinguish between two non-trivial cases: $x\in (L_1\setminus L_2)$ and $x\in L_2\setminus L_1$. In these cases, membership-certificates for $L_1$ would be nonmembership-certificates for $L_2$ and vice versa. Hence, you always have nonmembership-certificates for every input string $x$. And conclude that both $L_1$ and $L_2$ are in $\mathrm{co}$-$\mathrm{NP}$.