The striker will now remain at Ibrox until at least the end of the season.
"Obviously when I heard I had the opportunity to move to the Premiership and the offer was right both for myself and Rangers, it was difficult to turn down," said Cousin.
"However, I made it clear all along that I have enjoyed my time at Rangers and I am more than happy to stay, fight for my place and help the manager and the rest of the squad compete for silverware this season."