= Solution
The <Seifert-van Kampen theorem> says that if a path-connected space is the union of path-connected open sets $U,V$ with path-connected intersection, then
$$
\pi_1(U\cup V)=\pi_1(U)*\pi_1(V)/\langle\!\langle i_{U*}(w)i_{V*}(w)^{-1}:w\in\pi_1(U\cap V)\rangle\!\rangle.
$$
Using the usual polygonal cell decompositions, the torus attaches a two-cell to $S^1\vee S^1$ along $aba^{-1}b^{-1}$, while the projective plane attaches one along $a^2$. Hence
$$
\boxed{\pi_1(T^2)=\langle a,b\mid aba^{-1}b^{-1}\rangle\cong\mathbb Z^2},
\qquad
\boxed{\pi_1(\mathbb {RP}^2)=\langle a\mid a^2\rangle\cong\mathbb Z/2}.
$$
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