Cozart earned first-team all-state honors after leading Bishop Miege High School to the Kansas Class 5A State Championship game in 2012. Cozart threw for 2,759 yards and 25 touchdowns as a senior at Bishop Miege HS where he was coached by Jon Holmes.
Rated as a 3-star quarterback by Rivals.com, Cozart chose to attend the University of Kansas and join its football program over offers from Indiana, Kansas State, Minnesota, Northern Illinois and West Virginia. At Kansas, Cozart suffered a season-ending injury four games into his junior year; the NCAA granted Cozart an extra year of eligibility due to his injury. Cozart saw the most action in his fourth year with the Jayhawks, completing 112 of 190 pass attempts for 1,075 yards with 7 touchdowns and 9 interceptions. On February 21, 2017, Cozart announced he was leaving Kansas to seek a graduate transfer and would be eligible during the 2017 season. Cozart announced on May 4, 2017, that he will join the Boise State Broncos football team.