Wow, he's pretty! Probably a half-moon tail, but he might be a rosetail--there's a lot of branching in his tail.
Can you get a picture of him flaring? Just show him a mirror or a picture of another betta. Or turn all the lights off in the room he's in except the tank light. Sometimes showing them colorful objects works as well. Mine flares at peacock feathers.
Does he have a name yet?