So I supposed $|H \cap K|>1$
$\Rightarrow |HK||H \cap K|> |HK|$
Which eventually implied that
$\Rightarrow |H \cap K|>|G|$
Thus since G is a group, and H and K are subgroups then the identity belongs to both H and K. Since it belongs to both H and K then it belongs to $H \cap K$. But if $|H \cap K|>|G|$ then that implies $|H \cap K|$ is greater than one meaning it has at least two elements.
Im not sure if thats the outcome I should get but I was wondering if there are other ways to prove this problem.