= Solution
Choose distinct nonidentity elements $g_1,g_2\in G$ and $h_1,h_2\in H$, and put
$$
u=g_1h_1,
\qquad
v=g_2h_2.
$$
Expand a freely reduced word in $u^{\pm1},v^{\pm1}$. At a boundary where two syllables from one factor meet, their product is one of $g_1^{-1}g_2$, $g_2^{-1}g_1$, $h_1h_2^{-1}$, or $h_2h_1^{-1}$, all nonidentity by the choices above. Every other boundary already alternates between the factors. Thus the expansion reduces to a nonempty reduced word in $G*H$ and cannot represent the identity. The homomorphism from the rank-two <free group> sending its free generators to $u,v$ is injective, so
$$
\boxed{\langle u,v\rangle\cong F_2.}
$$
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