Why is it "jaayeengii" and not "jaaeengii"? It somehow makes sense to me to insert a consonant between the two vowels for reasons of easier pronunciation. But I looked up the rules somewhere else and there was no mentioning of that inserted "ya". Or is both possible again?
जाएँगी and जायेंगी are both accepted spellings.
Fun fact: the latter was used originally, and has historical significance. Sanskrit rules for spelling are extremely strict and according to them, no two vowels can be directly next to each other. This is why words like these required the य in between the two vowels. However, after India’s independence, Hindi underwent several spelling reforms. These include omitting the य for ease in practically understanding a word’s pronunciation. This gave rise to a class of words spelt like जाएँगी. This course accepts both the versions since both of them have wide usage.