Since we’re talking specifically about cougars, I can relay my experience shooting a wild mountain lion in Utah a decade ago. I went down there specifically in hopes of seeing the cat, which had been showing up at a couple of natural springs on an acquaintance’s property. The area was also frequented by black bears.
I sat in a tree stand above the spring, and had bear spray on me as my first line of defense. The property owner was in a ground blind further away and had his bear spray and a gun. I am not a gun owner, but I believe something like bear spray is a good idea and possibly more effective in tight quarters–obviously when pursuing bear photo ops but possibly with other animals–since even with the most careful, thoughtful approach in a large predator/animal encounter things can get unpredictable.
In our case, the other guy heard a bear in the brush and undid the safety on his bear spray. He forgot to put it back on, and it discharged in the car later that morning. Now I know why the stuff works.
Oh, and I got the lion shots later that day. Since then I have had numerous encounters with large predators and other animals (lots of pumas/mountain lions, charged by elephants and gorillas, etc.)… the scariest situation of all involved a venomous snake that slithered out of the frame toward me! Anyway, I find it invaluable to be respectful of the animal and to constantly monitor its comfort level–not just my own–in order to obtain photos without applying undue pressure or creating an uncomfortable situation for both parties. It’s even more helpful to employ locals who have much more familiarity with animal behavior (e.g., bear guides in Alaska, my puma guides in Latin America). I often rely on their instincts and knowledge, and follow their lead in these situations.