Lord Rama is an incarnation of the Supreme Personality of Godhead. Goddess Durga being the external (Maya) potency of the Supreme Personality of Godhead, why did Lord Rama perform a Yagya of Goddess Durga before killing Ravana as mentioned in Ramayana?
Saga Agastya preached Aditya Hrudhaya Stotra to The Lord Rama and informs him to perform Navratri pooja and yagna to The Goddess Durga to defeat Ravana.
Reference from Tyagaraja keerthanalu "Manasa Sancharare"
Maanasa Sancharare Brahmani Maanasa Sancharare Madasikhi Pinchaalankruta Chikure Madasikhi Pinchaalankruta Chikure Mahaneeya Kapolajita Mukure Maanasa Sancharare E Sree Ramaneekucha Sree Ramaneekuncha Sree Ramaneekucha Durga Vihaari Sevakajana Mandira Mandhaare Paramahamsamukha Chandra Chakore. E Pari Poorita Muraleeravadhaare Maanasa Sancharare
Besides, Ram leela is observed on 10th day(Dushera) after navratri. Lord Rama killed the deamon Ravana after he performed navratri pooja.