Recently someone was killed by a cougar in Washington. Apparently while the cougar was busy attacking his cycling buddy, he attempted to flee. That triggered the chase and kill instinct of the cougar, which then left his buddy to go after him.

What are your tips for defending against cougars?

It appears that unlike for defending against bear attacks, you can no longer bring a .22 to shoot your buddy in the knee so you can flee.

Yeah, read about that one! I'm in Washington now but luckily not in cougar country. Probably awareness and a sidearm?

Do not turn your back if you see them. Do not run. Fight tooth and nail with anything you have.

This was the first cougar related fatality since 1924 in Washington. Running was evidently a mistake. Stand your ground and make yourself look as big as possible...

Here is a documentary on how it´s done ;):
https://www.youtube.com/watch?v=rpI_Foef7-o&t=70
Application in real life may be a tiny bit harder.
In news about victims that survived with fairly small injuries the stories had in common that the people fought back and the buddies attacked the cougar with all the could weaponize.

The bikers stood their ground, even using a bike as a weapon at first. This was a determined lion. Most attacks in WA end as soon as the person puts up any kind of resistance. One mom snatched her little one from the jaws of a cougar after pummeling it with a water bottle. Puma-ling.
http://www.khq.com/story/11079274/officers-tracking-cougar-that-reportedly-attacked-child

There has been an increase in Mt Lion attacks in Washington State in the last 20 years. When the lion population gets too dense, the young starving lions get more desperate. They speculate this one was so hungry it wouldn't give up. Some methods of lion hunting, and numbers allowed to be taken, have been reduced in recent years in WA state. Some think this may be having the effect of an increase in attacks. Others say less hunting makes for more stability.

Pepper spray has worked for one biologist we met presenting wildlife information a cub scout camp. On three occasions he said.

Yelling worked in this situation
http://www.spokesman.com/blogs/outdoors/...-while-camping/

Mt Lion hunters can legally use smaller caliber centerfire cartridges for taking than deer hunters can in WA state. 22 caliber for lion, 24 for deer minimum.

As I understand it, the cyclist who sped away (and was killed) did so to draw the animal away from his companion, whose head was in the cougar's jaws and being shaken side to side.

So, he was hardly "running away," but rather trying to help. Tragically, his ploy worked -- at a terrible cost. I wasn't there, so I'm not inclined to armchair quarterback. Coming across a cougar that aggressive and persistent in broad daylight is the equivalent of a UFO.

Aren't bears the bears of the woods?

Just for perspective, how many have died as the result of encounters with dogs? honey bees? various insects??

Be alert for mountain lions,for sure, but they aren't the only hazard out there. Statistically, they are trivial.

I have encountered exactly 2 cougars in all my years of tramping around the outdoors.

The first one was way too close for comfort and left me spooked for quite some time. The second was a bit further away and it was a privilege to see such a magnificent animal for a few long seconds before it bounded away.

As HM mentioned, plenty of other threats out there that will take you down and snuff out your life in an instance, before an extremely rare chance of a cougar attack ever will.