Proof of equality with destructuring let...in
You can destruct not only on variables, but on expressions like f n as well. So, this will get you a solution:

Lemma foo : forall n, f n = (let (x, y) := f n in (x, y)).
  intro n.
  now destruct (f n).    (* or destruct (f n). reflexivity. *)
Qed.