9. (16 points) Let $n$ be an odd number $(n>1)$. The $n$ roots of the equation $z^{n}=1$ are $1, x_{1}, x_{2}, \cdots, x_{n-1}$. Find the value of the expression $\sum_{i=1}^{n-1} \frac{1}{1+x_{i}}$.
9. Let $y_{i}=\frac{1}{1+x_{i}}(i=1,2, \cdots, n-1)$.
Then $\left(\frac{1}{y_{i}}-1\right)^{n}=1 \Rightarrow\left(1-y_{i}\right)^{n}=y_{i}^{n}$.
Notice that, $n$ is odd.
Thus, $y_{i}^{n}+\left(y_{i}-1\right)^{n}=0$.
Therefore, the $n$ roots of the $n$-degree equation $2 y^{n}-n y^{n-1}-\cdots-1=0$ are $\frac{1}{2}, y_{1}, y_{2}, \cdots, y_{n-1}$.
By Vieta's formulas, we have
$$
\begin{array}{l}
\frac{1}{2}+y_{1}+y_{2}+\cdots+y_{n-1}=\frac{n}{2} \\
\Rightarrow \sum_{i=1}^{n-1} \frac{1}{1+x_{i}}=y_{1}+y_{2}+\cdots+y_{n-1}=\frac{n-1}{2} .
\end{array}
$$