If you do something cockily, you do it with too much confidence — the kind that makes other people tired of you. You walk in as if you have already won; you talk as if you cannot be wrong. It is not a compliment. It is close to arrogantly, though actually cockily is lighter: more swagger and showing off than cold pride. It is informal. You will hear it in speech and in stories more than in careful writing. Teachers and friends use it when someone is getting ahead of themselves.