The singer, who won the seventh run of the Fox competition, has recorded a new version of 'Don't You (Forget About Me)', originally released by Simple Minds.
The song, which was chosen by Idol creator and executive producer Simon Fuller, will play each Thursday following an elimination.
Cook is currently working on the follow-up to his debut album, which is scheduled for release through 19 Recordings/RCA Records later this year. His rendition of 'Don't You (Forget About Me)' is available for free download now to fans who pre-order the album on his official website, and will be released for general download on March 8.
This week saw the Top 13 contestants for American Idol's tenth season revealed. The public voted through five girls and five boys, while judges Jennifer Lopez, Randy Jackson and Steven Tyler added a further three wildcard picks.
American Idol airs Wednesdays and Thursdays on Fox.