Is it rude to get someone admit that they have lied to you by tricking them?
For example, asking them questions about the incident and waiting until they mess up an important detail.
Ex: My grandmother was sent to the hospital after a fall, but my dad suspected that she had put herself there on purpose to get sympathy and hadn't actually fallen by accident. He questioned her about how she had fallen, and caught her in a lie when she couldn't remember whether she was supposed to tell him or the hospital staff that she had hit her head when she fell. (She hadn't hit her head, but wanted to tell my dad that she had so that he would worry more. She didn't tell that to the hospital staff because she didn't want to actually be kept in the hospital).
Being truthful, yet misleading:
Person 1: "So you are the CEO's best friend? Isn't Alex a great guy?"
Person 2: "Yeah, he is! So since I'm such great friend of the CEO, could you do me that favor please?"
Person 1: "Alex is a woman."
Or even outright lying to them:
Person 1: "I hate Bob! You hate Bob too, right? Isn't he such a jerk? I wish someone would teach him a lesson!"
Person 2: "Then you'll appreciate this. I just stole all the cookies from the cookie jar and framed Bob!"
Person 1: "Well I'm actually Bob's best friend, so I'll be sure to tell everyone what you just said to me."
Are any of these approaches to get the truth rude? I'm talking only about things where the lie actually matters and could cause a problem. Not something stupid and none of your business like suspecting that your friend went to the adult store when they actually said they were getting groceries and trying to get them to admit where they really were.