If you are properly licenced, and are merely borrowing your cousin's car, with his permission, you are covered (unless you are, for some reason a prohibited driver on the policy) by his insurance. Insurance goes with the car. I drive other cars regularly, and if I loan my car to someone else, he/she is covered by my policy.
If you are taking the car home, in possession of it for more than 30 days (15 in some states, he needs to check his policy) that is a different story.
If the only insurance is liability, though, that is not car insurance, it doesn't cover any damage or loss to the vehicle, it covers damages done to other persons and their property in an at-fault accident.