"The cow likes the mouse."
Translation:Die Kuh mag die Maus.
This is a lovely sentence at first glance, but what it could actually mean is "The cow likes the mouse for dinner."
If by "per Duolingo", you mean that those two forms are both present in the translation hints for "der", please remember that the hints are not sentence-specific and may contain translations that are not appropriate in the sentence you are currently translating.
der may be:
- masculine nominative
- feminine genitive
- feminine dative
- plural genitive
However, here we have two feminine nouns (Kuh, Maus), one in the nominative case (the subject Kuh) and one in the accusative case (the direct object Maus).
Thus we need the feminine nominative article die before Kuh and the feminine accusative article die before Maus.
der is not appropriate anywhere in this sentence.