5. Let $S=\left\{(x, y) \mid x^{2}-y^{2}=\right.$ odd, $\left.x, y \in R\right\}, T=\{(x, y) \mid$ $\left.\sin \left(2 \pi x^{2}\right)-\sin \left(2 \pi y^{2}\right)=\cos \left(2 \pi x^{2}\right)-\cos \left(2 \pi y^{2}\right), x, y \in R\right\}$. Then
(A) $S \subset T$;
(B) $T \subset S$;
(C) $S=T$;
(D) $S \cap T=\Phi$.
5. (A)

When $y^{2}=x^{2}+$ an odd number, it is easy to see that
$$
\sin \left(2 \pi x^{2}\right)-\sin \left(2 \pi y^{2}\right)=\cos \left(2 \pi x^{2}\right)-\cos \left(2 \pi y^{2}\right)
$$

holds.
Therefore, when $(x, y) \in S$, it must belong to $T$, so $S \subseteq T$. Moreover, points $(x, y) \in T$ satisfying $x=y$ do not belong to $S$. Hence, $S \subset T$.