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

We prove that the function \(g(x)= 1 / ( 1 - \cos(x) )\) 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

$$ \lambda_n \leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \quad (n \geq 0 \ \mbox{even}) $$

and

$$ \mu_n \leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \quad (n \geq 1 \ \mbox{odd}) $$

hold for all \(x,y\in (0,\pi)\) with \(x+y\leq \pi\).