Let \(\lambda_ 1,\lambda_ 2,..\). be the positive roots of the equation tg x\(=x\). The following recurrence formula \[ \sum^{n-1}_{k=1}(- 1)^{k-1}k((2k+1)!)^{-1}S_{2n-2k}=(-1)^ nn(n-1)((2n+1)!)^{-1} \] for all \(n\geq 2\), and \(S_ 2=0.1\) for the sums of the series \(S_{2n}=\sum^{\infty}_{k=1}\lambda_ k^{-2}\) are proved. This result, from the infinity product for the entire function sin x-x cos x, is obtained. The sums of the series \[ \sum^{\infty}_{k=1}(- 1)^{k+1}\lambda_ k^{-2n}(1+\lambda^ 2_ k)^{-1/2}\quad for\quad n=1,2,3 \] are calculated, too. Moreover, the analogous problems for the equation cotg x\(=x\) are considered.