In Bott&Tu's book, it said that:
If a 1-cocycle $\eta=(\eta_{01},\eta_{02},\eta_{12})$ is a coboundary, then $\eta_{01}-\eta_{02}+\eta_{12}=0$. So $\eta=(1,0,0)$ is a non-trivial 1-cocycle on the circle.
But the equation certainly does not hold when $\eta=(1,0,0)$. What's wrong? Any hint would help.
The statement is that if you have a 1-cocyle $\eta$ which happens to be a coboundary, then it satisfies the equation. The cocycle $(1,0,0)$ does not satisfy the equation, therefore it is not a coboundary. This is what they mean when they say "non-trivial cocycle": a cocycle which is not a coboundary.