One reason your cousin always has to be right (at least in his own eyes) may be pride. In other words, it would be a blow to his pride to admit he's wrong -- and therefore he refuses to do it.
But pride is a dangerous and deceptive sin (and that's what it is -- a sin). For one thing, it cuts us off from other people. No one likes someone who's arrogant and prideful all the time, because they always act as if they're better than anyone else. And pointing out the faults we see in others while refusing to admit our own faults doesn't win us any friends, either. The Bible warns, "Pride only breeds quarrels, but wisdom is found in those who take advice" (Proverbs 13:10).
I'm not a psychologist, of course, but I've often observed that people like this actually may be very insecure and lacking in self-confidence. Their pride is like a wall that they hope will keep people from discovering what they're really like. Tragically, their pride also keeps them from God.
But it doesn't need to be that way, because God knows all about us anyway -- and yet He still loves us. He loves us so much that He sent His Son into the world to redeem us and make us part of His family forever. Pray for your cousin, and urge him to discover the joy and peace that come only from having Christ in our hearts.