There are many synonyms for the phrase "aggressively self-confident." Some examples are cocksure, conceited, arrogant, overconfident, and haughty. A cocksure person is very confident but sometimes can be seen as arrogant. Conceited means to have an excessively high opinion of oneself. Arrogant means to have an exaggerated sense of one's abilities or importance. Overconfident means to have an unwarranted sense of confidence. Haughty means to be disdainful and aloof. While these words all describe someone who is self-assured, they also carry different connotations. It's important to choose the right word that accurately reflects the person's behavior and personality.