Example 4 Solve the equation $(x+m)^{n}-(x-m)^{n}=0$.

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Solution: Clearly, $x=m \neq 0$ is not a root of the original equation. Let $y=\frac{x+m}{x-m}$. Then, when $m \neq 0$, $y \neq 1$, and $x=\frac{m(y+1)}{y-1}$. Thus, the original equation is transformed into $y^{n}=1$, whose roots different from 1 are
$$
y_{k}=\cos \frac{2 k \pi}{n}+i \sin \frac{2 k \pi}{n}, k=1,2, \cdots, n-1 \text {. }
$$

Therefore, $x_{k}=\frac{m\left(y_{k}+1\right)}{y_{k}-1}=m \cdot \frac{i \sin \frac{2 k \pi}{n}+\left(1+\cos \frac{2 k \pi}{n}\right)}{i \sin \frac{2 k \pi}{n}-\left(1-\cos \frac{2 k \pi}{n}\right)}$
$$
=-m \cdot \cot \frac{k \pi}{n} \cdot \mathrm{i}(k=1,2, \cdots, n-1)
$$

These are the solutions sought.