Connect QO (the angle to be proved is a right angle) with the center o, ∠QAP=∠AQO.

∠QOP=∠OPQ

∠QAP+∠AQO+∠QOP+∠OPQ = 2(∠QAP+∠QPA)= 180

(∠QAP+∠QPA)=90

∠AQP=90