Yes, you are entitled to the AOTC for your dependent. Please see below for more detailed explanation from the IRS.
With regard to student loans: expenses paid with borrowed funds are "deemed paid" by the borrower. Even payments made with a credit card would count. The source of funds is not important; it is the act of paying the tuition and fees during the tax year that matters, even if you have incurred a debt to a third party.
The following guidance for the American Opportunity Tax
Credit is taken from IRS Pub. 970 Tax Benefits for
Education. Identical provisions exist for the
Lifetime Learning Credit.
"If there are qualified education expenses for your
dependent during a tax year, either you or
your dependent, but not both of you, can claim an American Opportunity
Credit for your dependent's expenses for that year.
For you to claim an American Opportunity Credit for your
dependent's expenses, you must also claim an exemption for your dependent. You
do this by listing your dependent's name and other required information on Form
1040 (or Form 1040A), line 6c.
Expenses paid by dependent. If
you claim an exemption on your tax return for an eligible student who is your
dependent, treat any expenses paid (or deemed paid) by your dependent as if you
had paid them. Include these expenses when figuring the amount of your American
Opportunity Credit.
Expenses paid by you. If
you claim an exemption for a dependent who is an eligible student, only you can
include any expenses you paid when figuring the amount of the American
Opportunity Credit. If neither you nor anyone else claims an exemption for the
dependent, only the dependent can include any expenses you paid when figuring
the American Opportunity Credit."
