【Example 3.41】Solve the equation
$$
\mathrm{C}_{2 n+1}^{1} x^{n}-\mathrm{C}_{2 n+1}^{3} x^{n-1}+\mathrm{C}_{2 n+1}^{5} x^{n-2}-\cdots+(-1)^{n}=0
$$
Let $x=\cot ^{2} \alpha$. From Example 1.12, we know that
$$
\begin{array}{l}
\sin (2 n+1) \alpha \\
\quad=\cos ^{2 n+1} \alpha\left(\mathrm{C}_{2 n+1}^{1} \tan \alpha-\mathrm{C}_{2 n+1}^{3} \tan ^{3} \alpha+\cdots\right) \\
\quad=\cos ^{2 n+1} \alpha \tan ^{2 n+1} \alpha\left(\mathrm{C}_{2 n+1}^{1} \cot ^{2 n} \alpha-\mathrm{C}_{2 n+1}^{3} \cot ^{2 n-2} \alpha+\cdots\right)
\end{array}
$$

Since when $\alpha=\frac{\pi}{2 n+1}, \frac{2 \pi}{2 n+1}, \cdots, \frac{n \pi}{2 n+1}$, $\sin (2 n+1) \alpha=0$, it follows that $\cot ^{2} \frac{\pi}{2 n+1}, \cot ^{2} \frac{2 \pi}{2 n+1}, \cdots, \cot ^{2} \frac{n \pi}{2 n+1}$ are solutions to the original equation.