The 20-year-old star was a guest on The Tonight Show With Jay Leno along with the US stand-up comedian.
After his interview, Cook, 38, stripped off his shirt to reveal a tattoo of Swift's face on his chest and he then performed a song for her with his guitar after admitting he was a big fan.
His lyrics included: 'Taylor Swift, you're a gift. Give me a lift to your heart, beating inside my chest, I did my best. Download it now.
'Now back to you, Taylor Swift, stand up girl, take a bow. Show us that smile, don't be shy, I hope her fans encourage her now. Come on everybody!
'We love you Taylor, Taylor, Taylor Swift!'
He also had the words 'I Did My Best' tattooed on his back, the words of his new greatest hits album.
The country singer then got up and took a bow as the audience applauded him for his performance.