  We prove that the function $g(x)= 1 / \bigl( 1 - \cos(x) \bigr)$ 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
$x+y\leq \pi$.
