You don't need to "prove it", when you file your tax return. You claim the credit on form 8863, then lines 50 & 68 of form 1040, based on your own records.

If you have an IRS letter denying the credit, you need to explain the circumstances. Generally, you need to show you qualify. See https://www.irs.gov/Individuals/AOTC . Or they may just be asking for proof you paid. The 1098-T will frequently only indicate what you were billed, not what you actually paid.

For further discussion, see https://ttlc.intuit.com/questions/2956076-why-am-i-receiving-0-return-for-american-opportunity-credit-i-meet-all-pre...