I prefer animals to people - there is not an animal or insect I am averse to. As mentioned in previous posts, I have two big strong, muscular boxers. When I take them out walking, and a cat nears them, they don't bother the feline, but it stops in it's tracks, and stares at them, as if saying 'come on - if you're hard enough!' Now, if my dogs attacked the cat, they would do it serious damage, but they are not vicious natured, and wouldn't touch it. If the cat came near them, they'd turn and run.
Each time this occurs, it is so amusing. How hard faced are cats? Do they not realise how dangerous it could be?

I can only assume the cat has utmost confidence in its' ability to outrun the dog
To be fair most cats can outrun most dogs - though they may find out the hard way that that doesn't apply to greyhounds

Where I used to live there was one cat who was so confident around dogs, it used to actually take a swipe at them if it got near enough.
It was really irritating, because my dog would certainly have a go and I'd end up in trouble if she hurt it, but this damn thing was always hanging around.

It disappeared one day, probably picked on the wrong dog lol

Because that is how cats try and assess a threat situation.

When a cat realises it is being watched or stared at, it may stop whatever it is doing, assess the "threat" and then continue with its activity, but in a far more self-conscious way. The cat knows it is being watched and becomes uncomfortable. Only when it is no longer being watched, does it relax again.

So by stopping and staring at your dogs it is keeping an eye on the "threat", so if your dogs do go for him/her they know where to run (i.e in the opposite to where your dogs are)

So it isn't being cocky or brazen, it's just being a cat