While this is an important skill to have, I've also learned over time that there are certain types of people where it's easier to let them "be right" and accommodate the consequences. I didn't always think this way. In the past, I would say that people are reasonable, it's always possible to figure out what the truth is, especially if I acknowledge the possibility that I am often wrong. But now I realize that some people have a very skewed view of the world around them, either built up from a lifetime of emotional trauma, undesirable upbringing, or something else. edit: it is unfortunate when their own skewed perspective causes self-infliction of additional emotional trauma, it's a very negative and constant feedback loop. And that skewed perspective causes any and all disagreements to be perceived as personal attacks or worse. Every time I feel like I make progress with these people to earn their trust enough to get them to talk with me, it's back to square 1 when a new issue crops up, and then it's once again obviously because the world is out to get them, and there is no possible other explanation.
Eventually, I get to a crossroad: either I accept what these people are saying as right (even when it's wrong and would require some absurd actions on my part), or I navigate to exit that situation so I can minimize my future interactions with those people. If neither are possible, it's perhaps time to find new work. Of course, I'd love to get onto the same page as them. But I think we have to acknowledge that sometimes it's not possible. I say this as someone who believes that he has gotten pretty good at this skill after years of experience.
As would probably be expected, these types of people are lifers at organizations where they have used political means to entrench themselves, and would likely not survive on the open market if they ever needed to look for new employment.