Inequalities for 1/(1-cos(x)) and its derivatives

We prove that the function $g(x)=1\mathrm{/}(1-\cos (x))$ is completely monotonic on $(0,\pi ]$ and absolutely monotonic on $[\pi ,2\pi )$, and we determine the best possible bounds ${\lambda }_n$ and ${\mu }_n$ such that the inequalities $${\lambda }_n\le g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y)(n\ge 0even)$$ and $${\mu }_n\le g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y)(n\ge 1odd)$$ hold for all $x,y\in (0,\pi )$ with $x+y\le \pi$.