On page 120 of Bott and Tu, it is shown that the Euler class obtained from the de Rham Čech complex and that obtained by direct construction via the structure group of sphere bundles agree. In the course of that proof a homoptopy identity is used:
$$\delta K+ K \delta= 1$$
For a cocycle $\xi$ of course this means $\delta K \xi =\xi$ but it looks like they are using some property such that actually $K\delta \xi=\xi.$ In this case $\xi$ is the 1-form constructed in the course of obtaining the global angular form. It is in the middle of page 120. I do not get why that is.
pp 120, line ~ 12 "We may take ξ to be 1/(2π)Kdϕ." Is the crux of the matter. I do not see why.
The cochain $\xi$ is defined by the property that $\delta\xi=\frac{1}{2\pi}d\phi$, and any other cochain satisfying this equation can be used to define the Euler class in place of $\xi$ (cf. the Claim at the end of page 72). Now note that $\xi=\frac{1}{2\pi}Kd\phi$ satisfies $$\delta\xi=\frac{1}{2\pi}\delta Kd\phi=\frac{1}{2\pi}(1-K\delta)d\phi=\frac{1}{2\pi}d\phi-K\delta d\phi=\frac{1}{2\pi}d\phi,$$ with the last equality being because $d\phi$ is a cocycle so $\delta d\phi=0$. (This calculation is essentially what the second paragraph on page 95 is talking about, with $d\phi$ in place of what is called $\phi$ there.) Thus we can take $\xi=\frac{1}{2\pi}Kd\phi$.