In order to claim the tuition credit, the student must be your dependent. So, in your case, you claim the tuition credit on your dependent. The other parent claims the tuition credit on his/her dependent. You each get to count the tuition paid by the other parent, in calculating your credit. You also get to count the third third (I assume paid by the student).
Although the general rule, in taxes, is that you must be the one making the payment, to get the deduction or credit, there is an exception for education.
Room & board are not qualified expenses for claiming the tuition credit.
