5. Let $S=\left\{(x, y) \mid x^{2}-y^{2}=\right.$ odd, $\left.x, y \in \mathbf{R}\right\}, T=\left\{(x, y) \mid \sin \left(2 \pi x^{2}\right)-\sin \left(2 \pi y^{2}\right)=\right.$ $\left.\cos \left(2 \pi x^{2}\right)-\cos \left(2 \pi y^{2}\right), x, y \in \mathbf{R}\right\}$, then
A. $S \varsubsetneqq T$
B. $T \varsubsetneqq S$
C. $S=T$
D. $S \bigcap T=\varnothing$
$$
\begin{array}{l}
\sin \left(2 \pi x^{2}\right)-\sin \left(2 \pi y^{2}\right)=\cos \left(2 \pi x^{2}\right)-\cos \left(2 \pi y^{2}\right) \\
\Rightarrow 2 \cos \left(\pi x^{2}+\pi y^{2}\right) \sin \left(\pi x^{2}-\pi y^{2}\right)=-2 \sin \left(\pi x^{2}+\pi y^{2}\right) \sin \left(\pi x^{2}-\pi y^{2}\right),
\end{array}
$$

When $x^{2}-y^{2} \in \mathbf{Z}$, the above equation holds, so $S \varsubsetneqq T$. Therefore, the answer is $A$.