I asked my boyfriend about sharing passwords and his response was "lol no." What should I do?

So we've been together about six months, and I know the pin to his phone but haven't had a chance to go through it and make sure he's not talking to other girls behind my back, so today I texted him and asked him for his FB password and he refused, wanted to know why I want it and when I explained why he said "lol no, I'm not giving you my passwords just for that" and I told him how in relationships couples should share passwords but he won't budge and says I have two choices - have a relationship without his passwords, or not, what do I Say / do?

Updates
+1 y
Thanks all but one of you for not even being slightly helpful.

If he's not doing anything wrong why does it matter?
I asked my boyfriend about sharing passwords and his response was "lol no." What should I do?
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