Education

You don't have to prove anything when you file.  That comes later if you are audited. The 1098-T is only any informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or deduction or possibly your student has taxable scholarship income. 

If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)

You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T. In the 1098-T screen, click on the link "What if this is not what I paid the school" underneath box 1. You will then be able to enter the actual amounts paid.  You will also reach a screen that allows you to adjust the scholarship amount for "amounts not awarded for 2019 expenses".

That said, your own numbers indicate that almost all of the tuition was covered by grants.  But room &board, as well as books, computers  and other course materials are qualified expenses for the 1099-Q.

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Qualified Tuition Plans  (QTP 529 Plans)

It’s complicated.

For 529 plans, there is an “owner” (usually the parent), and a “beneficiary” (usually the student dependent). The "recipient" of the distribution can be either the owner or the beneficiary depending on who the money was sent to. When the money goes directly from the Qualified Tuition Plan (QTP) to the school, the student is the "recipient". The distribution will be reported on IRS form 1099-Q. 
The 1099-Q gets reported on the recipient's return.** The recipient's name & SS# will be on the 1099-Q.
Even though the 1099-Q is going on the student's return, the 1098-T should go on the parent's return, so you can claim the education credit. You can do this because he is your dependent.

You can and should claim the tuition credit before claiming the 529 plan earnings exclusion. The educational expenses he claims for the 1099-Q should be reduced by the amount of educational expenses you claim for the credit.
But be aware, you can not double dip. You cannot count the same tuition money, for the tuition credit,  that gets him an exclusion from the taxability of the earnings (interest) on the 529 plan. Since the credit is more generous; use as much of the tuition as is needed for the credit and the rest for the interest exclusion. Another special rule allows you to claim the tuition credit even though it was "his" money that paid the tuition.
In addition, there is another rule that says the 10% penalty is waived if he was unable to cover the 529 plan withdrawal with educational expenses either because he got scholarships or the expenses were used (by him or the parents) to claim the credits. He'll have to pay tax on the earnings, at his lower tax rate (subject to the “kiddie tax”), but not the penalty.

Total qualified expenses (including room & board) less amounts paid by scholarship less amounts used to claim the Tuition credit equals the amount you can use to claim the earnings exclusion on the 1099-Q. 
Example:
  $10,000 in educational expenses(including room & board)

   -$3000 paid by tax free scholarship***

   -$4000 used to claim the American Opportunity credit

 =$3000 Can be used against the 1099-Q (usually on the student’s return)

Box 1 of the 1099-Q is $5000

Box 2 is $600

3000/5000=60% of the earnings are tax free

60%x600= $360

You have $240 of taxable income (600-360)

**Alternatively; you can just not report the 1099-Q, at all, if your student-beneficiary has sufficient educational expenses, including room & board (even if he lives at home) to cover the distribution. You would still have to do the math to see if there were enough expenses left over for you to claim the tuition credit. Again, you cannot double dip!  When the box 1 amount on form 1099-Q is fully covered by expenses, TurboTax will enter nothing about the 1099-Q on the actual tax forms. But, it will prepare a 1099-Q worksheet for your records, in case of an IRS inquiry.

On form 1099-Q, instructions to the recipient reads: "Nontaxable distributions from CESAs and QTPs are not required to be reported on your income tax return. You must determine the taxability of any distribution." 

 ***Another alternative is have the student report some of his scholarship as taxable income, to free up some expenses for the 1099-Q and/or tuition credit.