Here the group $(H, K)$ is the group generated by elements of the form $hkh^{-1}k^{-1}$ with $h \in H$ and $k \in K$.
This is a question in Hungerford's Algebra.
It's pretty clear to me that we can show the desired result by showing that $H$ and $K$ are in the normalizer of $(H, K)$. However, it's not clear to me how to show this, that is, if $ c\in (H,K)$ why should $hch^{-1} \in (H, K)$. The trick used when showing that the commutator subgroup is normal fails as $hkh^{-1}$ need not be in $K$. It seems that other tricks used to show that the commutator subgroup is normal will fail as well considering that if $H$ and $K$ are not both abelian then $(H, K)$ is not the commutator of their join, and so isn't a typical commutator subgroup.
Any hints would be welcome!