Problem 874

330. a) The sum of the roots of the equation of degree $n$

$$x^{n}-\frac{C_{2 n+1}^{3}}{C_{2 n+1}^{1}} x^{n-1}+\frac{C_{2 n+1}^{5}}{C_{2 n+1}^{1}} x^{n-2}-\ldots=0$$

(see the solution of problem 229 b)) is equal to the coefficient of $x^{n-1}$, taken with the opposite sign, i.e.

$\operatorname{ctg}^{2} \frac{\pi}{2 n+1}+\operatorname{ctg}^{2} \frac{2 \pi}{2 n+1}+\operatorname{ctg}^{2} \frac{3 \pi}{2 n+1}+\ldots+\operatorname{ctg}^{2} \frac{n \pi}{2 n+1}=$

$$=\frac{C_{2 n+1}^{3}}{C_{2 n+1}^{1}}=\frac{n(2 n-1)}{3}$$

b) Since $\operatorname{cosec}^{2} \alpha=\operatorname{ctg}^{2} \alpha+1$, it follows from the formula of problem a) that

$\operatorname{cosec}^{2} \frac{\pi}{2 n+1}+\operatorname{cosec}^{2} \frac{2 \pi}{2 n+1}+\operatorname{cosec}^{2} \frac{3 \pi}{2 n+1}+\ldots$

$$\ldots+\operatorname{cosec}^{2} \frac{n \pi}{2 n+1}=\frac{n(2 n-1)}{3}+n=\frac{2 n(n+1)}{3}$$