More Jordan type inequalities

The function $\tan(\pi x / 2) / (\pi x / 2)$ is expanded into a Laurent series of $1 - x^2$, where the coefficients are given explicitly as combinations of zeta function of even integers. This is used to achieve a sequence of upper and lower bounds which are very precise even at the poles $x = 1, -1$. Similar results are obtained for other trigonometric functions with poles.