Short answer:  you are not allowed to do that. 

Longer answer: it's possible but not likely.

 Yes, you can say they paid the tuition. It doesn't matter who  paid the tuition. A taxpayer that  is allowed to claim the credit, may do so, regardless of who actually paid the tuition. Typically this means the parent may claim the credit, even it the student paid tuition with student loans.

The tuition credit goes with the student's exemption. But there is a rule that says IF somebody else CAN claim him as a dependent, he is not allowed to claim his own exemption. 

Technically, there is a special  provision that allows your daughter or son to claim a tuition credit. But, from a practical matter it seldom works out.  A student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he/she supports himself by working . She cannot be supporting herself on student loans & grants and 529 plans and parental support.  It is usually best if the parent claims that credit.  

If the student actually has a tax liability, she can claim a non-refundable credit but then the parent must forgo claiming the student's exemption and the student still can’t claim his own exemption. This special combination of circumstances just doesn't happen very often and even when it does, the math usually doesn't work out.