No, you don't. But...
The first key point is that if your child can be claimed, they must check the box saying they can be claimed, if you aren't going to claim them. This makes them ineligible for a number of tax benefits. So most of the time, it is better for you to claim them because then at least someone gets a benefit. If they don't check that box, as if they were independent, they might qualify for more financial aid and tuition credits, but if audited or discovered, they might have to pay it all back plus be barred from future claims.
The second point is that, specifically regarding the American Opportunity Credit, there is a circumstance where, if your income is too high to claim the credit, you can decide to not claim your child as a dependent, and your child can claim a partial AOTC credit. They would check "yes" for the question "can someone claim you as a dependent" but then would answer "no" to the followup question, "will that person actually claim you this year." The AOTC is that case is limited by several factors but some credit may be better than no credit, if you are income-limiteded out of the credit yourself.
If you are not income-limited out of the credit, there is very little reason not to claim your child as a dependent if he or she qualifies.
