He might have torn it on one of your plants or decorations. What kinds of decorations do you have in your tank? Betta fins are very sensitive so he might have scratched it on something.
For now, I recommend going to the store and adding some API StressCoat to your tank. It will help as your betta's tail repairs. If you notice the tear getting bigger, or tears in other places, your betta may have fin rot.
Filling out this form will help us help you:
What size is your tank?
What temperature is your tank?
Does your tank have a filter?
Does your tank have an air stone or other type of aeration?
Is your tank heated?
What tank mates does your betta fish live with?
What type of food do you feed your betta fish?
How often do you feed your betta fish?
How often do you perform a water change?
What percentage of the water do you change when you perform a water change?
What type of additives do you add to the water when you perform a water change?
Have you tested your water? If so, what are the following parameters?
Symptoms and Treatment
How has your betta fish's appearance changed?
How has your betta fish's behavior changed?
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Does your fish have any history of being ill?
How old is your fish (approximately)?