Relationship of $ lambda_ {max} (A ^ TA) $ and $ lambda_ {max} (AA ^ T) $

I try to show that:
$$ cond_2 (AA ^ T) = (cond_2 (A)) ^ 2 $$
I have already shown that
$$ (cond_2 (A)) ^ 2 = frac { lambda_ {max} (A ^ TA)} { lambda_ {min} (A ^ TA)} $$

But something trivial
$$ cond_2 (AA ^ T) = frac { lambda_ {max} (AA ^ T)} { lambda_ {min} (AA ^ T)} $$

Is there a way to reconcile them? Is it just a typo in the exercise?