Just man up and ask your brother "Are you still with (sister in law here), are you seeing someone else?"
Yeah, just ask him. Don't give out his password. Maybe if you know it you should check (have you ever snooped before on him, and why do you know his pass?).
No, I don't snoop on my brother. I know his password because we exchanged them. Sometimes he knows someone on fb he wants to show to me and the only way to view their full profile is through his. I never log in unless he gives me permission.
So while I'm tempted to log in and just see if he's contacting this girl I just love my brother for trusting me with his password in the first place and I would never, ever want to betray him.