Cook had been with the Bengals since 2007. He played in 73 games, starting 66. Offensive line coach Paul Alexander said the team hasn't decided who will start at center next season.
Tate would have become an unrestricted free agent if he didn't get a deal on Tuesday. His biggest impact has been on special teams during his three seasons with Cincinnati.
Tate is averaging 24.8 yards on kickoff returns, a franchise record. His average of 9.9 yards per punt return ranks second in team records. He is in position to set franchise records for punt and kickoff return yards next season.
AP NFL website: http://www.pro32.ap.org