I quote here a relevant passage in the Qur'an:
And do not insult those they invoke other than Allah, lest they insult Allah in enmity without knowledge...
Here, Allah clearly dissuades Muslims from a particular course of action due to the risk of it causing others to sin in retaliation. This implies that you as the instigator could still bear some responsibility for his actions, especially if you could have reasonably expected what sins (if any) his response would involve.
So do you bear responsibility in this particular case? Only Allah and maybe your cousin know the answer to that with any certainty; it could well be that he would've cursed regardless of anything you did. However, it is good that you are worried about this possibility, and you should keep it in mind next time you lose your temper.