= Solution
Use the amalgam from part (c), with edge group
$$
C=\langle a^2,c^2\rangle.
$$
An odd power of $a$ belongs to $A\setminus C$, because $C$ is the index-two translation subgroup of the <Klein bottle group> $A$. Similarly, an odd power of $c$ belongs to $B\setminus C$. Thus
$$
a^{i_1}c^{j_1}a^{i_2}\cdots c^{j_{n-1}}a^{i_n}c^{j_n}
$$
is a reduced alternating word whose syllables lie in $A\setminus C$ and $B\setminus C$. The <normal form theorem for an amalgamated free product> says that every nonempty reduced alternating word is nonidentity. The displayed element is therefore nontrivial for every $n\geq1$.
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