As long as your cousin is willing to ask questions, you should be willing to answer them -- even if you don't seem to be making any impact. After all, if you ignore his questions, he may conclude that you don't have any answers, and that he's right and you are wrong. The Bible says, "Always be prepared to give an answer to everyone who asks you to give the reason for the hope that you have" (1 Peter 3:15).
At the same time, you also may need to challenge your cousin by asking him if he's honestly seeking answers, or if he just wants to argue. Occasionally I've met people who wanted to ask me questions they knew were hard to answer, but in reality they weren't really interested in the answers.
Instead, they simply wanted to keep God at arm's length (so to speak), and their questions were a way of doing that. Someday, you might ask your cousin: If all your questions were answered, would you be willing to give your life to Christ?
The most important thing you can do, however, is to pray for your cousin. What should you pray? Pray first of all that he will realize he needs God. Pray, too, that God will convict him of his sin and his pride. Pray especially that God will turn his heart and mind to Christ. And pray for yourself, that you will be an example to him of Christ's love and truth.