They can trust it if the content is good. At the end having a good domain can help you get more traffic and also seem more legit (that's why I talked in favor of paid domains earlier) but if you have good and quality content people will trust you no matter what domain you choose.
True but if two sites with quality content on the topic you are searching for popped up in the search engines, one to a .com (or other well known gTLD) and the other to a .tk Which would you choose?
Another thing is that .tk is a ccTLD which means the content on it should be specific to that area. So if you are going to have a website with quality content for people searching for things in or around Tokelauan then go for it!
If your not then you might as well just invest a few buck on a good gTLD.