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I try to do my best to avoid conflicts with such coworkers, as a friendly atmosphere at a workplace leads to better productivity. I had several situations throughout my career when I had to deal with coworkers who were acting like a "boss" without being competent. There were a couple of things I did to work this through. However, only one thing worked best for me. I used to invite them for a cup of coffee to talk through the situation. I noticed a significant improvement after such conversations.
This is a common one, and honestly the answer is complicated because everyone's work environment is a little different. Currently I work in an environment where annual reviews are done by your co-workers and then sent to the manager for review. In this case, I would just remember the way I was treated come time to fill out that person's annual review. You could also just talk to your boss, and if that doesn't work bring it up with HR. If nothing else works, confront the person.
Prepare your facts. If you have to work together on a project, have your part ready, airtight, with no holes, and make sure to have supporting documents to uphold your ideas. And if you know you are right, and everyone else does too, just nod and smile at their outrageous claims.
Let his/her acting fail via presnting your idea with knowledge. Such co-worker could not continue it for long. soon they have to change their getup as supporter.