Ask him if he has it.
He will probably get mad you are invading his privacy and not trust you anymore since he does not do that with you. You should ask him if he is cheating on you and then say something like then if you trust me and want me to trust you let me see your emails. If he says no then he has something to hide. Because my friend her boyfriend lets her read his texts and emails now that are my relationship goals for the future.
Rather than trying to catch him online and trying to get his password, don't you think it would be best to just confront him about it? Just look into his eyes and you'll know if he's lying. Much better to just get it out in the open than sneaking around.
You can get someone's gmail password by trying to log into their account, pretend you forgot the password and then select the password reset option.
Then ask to use your friend's phone to do something (if he or she trusts you), and then open the email and reset the password to something you will know.
This will allow you to check their email later, at your own leisure.
HOWEVER... No matter whether your friend is cheating on you, getting his gmail password is not only a horrible thing to do, it could be considered illegal too.