Keep in mind that I studied this stuff 15 years or more ago, but at that time I learned a couple things that wouldn’t be apparent to the average person, but could be very useful.
First, police officers can lie to you. It’s perfectly legal, and indeed, a popular technique.
Second, if arrested, don’t merely say you’re invoking your right to remain silent - this theoretically allows police to come ask you questions again after a “reasonable” amount of time. Instead, say you want to talk to a lawyer. The rule (at the time I learned at least) is that they need to stop asking questions, period, at that point.
While I would definitely recommend saying little in general, and never trusting an LEO to be on your side, it’s also smart to be polite and cooperative to a certain degree. I got stopped recently in a random* driver’s license check, and the thought flitted across my mind whether I should cooperate, but it is true that you need a valid license to drive, and I’m not at all sure what my rights are to refuse that. So I cheerfully showed my license and went about my business. Now, if he had asked to search my vehicle, I would have said, “Respectfully, I do not give permission for you to search.” Don’t have anything to hide, have never even tried illegal drugs or transported anything contraband, but I have my civil libertarian line that won’t be crossed, and why expose yourself to even a slight risk? My friend allowed a search of her car after a small accident, and it turned out her hot mess of a sister had dropped a joint in the car. Better to just keep the police out of your business when you can.
In that recent case in Florida where a young woman was arrested for taking indecent liberties with a minor (her high school girlfriend), her father said she cooperatively talked with the LEOs because her dad is in law enforcement and she had learned to trust such people. In the process, she completely and unequivocally incriminated herself. What could have been a dodgy case for the prosecution to prove (they would only have a very hostile witness on their side) became a slam-dunk, and now she’s on the hook as a possible registered sex offender.
*I suspect they were looking for a reason to pull over some Latino landscapers who they probably suspected of not having valid licenses, but don’t know for sure.