Final answer:
Both the Ottoman and Mughal empires adopted policies of limited religious toleration to establish legitimacy among non-Muslim majorities, evidenced by Mehmed II's attitudes in the Ottoman Empire and Akbar the Great's policies in the Mughal Empire.
Step-by-step explanation:
The development in the Ottoman and Mughal empires in the Early Modern Period that best supports the author's claim that these empires succeeded in winning legitimacy from populations with non-Muslim majorities is that rulers in both empires adopted policies of limited religious toleration. This is seen in the actions of Mehmed II in the Ottoman Empire with the conquest of Constantinople, inviting scholars and artists from various backgrounds while tolerating his non-Muslim and European subjects, and in the Mughal Empire under the leadership of Akbar the Great who was known for his religious eclecticism and tolerance towards all faiths. These policies helped to provide stability and establish the legitimacy of their rule among diverse populations.