I don't know of Gary Cooper ever signing his name as Frank Cooper. However, this photograph is from a Frank Capra film and the autograph resembles other Capra examples. I think it's highly possible you have a Frank Capra autograph instead.
https://www.baumanrarebooks.com/BookImages/87668.jpg - Frank Capra just signing Frank
No problem! Frank Capra is an awesome autograph to have in itself if you are a movie fan.