Example 8 Find:
(1) $\mathrm{C}_{n}^{1}-3 \mathrm{C}_{n}^{3}+5 \mathrm{C}_{n}^{5}-7 \mathrm{C}_{n}^{7}+\cdots$;
(2) $2 C_{n}^{2}-4 C_{n}^{4}+6 C_{n}^{6}-8 C_{n}^{8}+\cdots$.
Solution: From the binomial expansion and (III), we have
$$
\begin{array}{rl}
n & n(1+\mathrm{i})^{n-1} \\
= & n\left(\mathrm{C}_{n-1}^{0}+\mathrm{i} \mathrm{C}_{n-1}^{1}+\mathrm{i}^{2} \mathrm{C}_{n-1}^{2} \cdots+\right. \\
& \left.\mathrm{i}^{n-1} \mathrm{C}_{n-1}^{n-1}\right) \\
= & n\left(\mathrm{C}_{n-1}^{0}-\mathrm{C}_{n-1}^{2}+\mathrm{C}_{n-1}^{4}-\cdots\right)+ \\
& \mathrm{i} \cdot n\left(\mathrm{C}_{n-1}^{1}-\mathrm{C}_{n-1}^{3}+\mathrm{C}_{n-1}^{5}-\cdots\right) \\
= & \left(\mathrm{C}_{n}^{1}-3 \mathrm{C}_{n}^{3}+5 \mathrm{C}_{n}^{5}-7 \mathrm{C}_{n}^{7}+\cdots\right)+ \\
& \mathrm{i}\left(2 \mathrm{C}_{n}^{2}-4 \mathrm{C}_{n}^{4}+6 \mathrm{C}_{n}^{6}-8 \mathrm{C}_{n}^{8}+\cdots\right) .
\end{array}
$$

On the other hand,
$$
\begin{array}{l}
n(1+\mathrm{i})^{n-1} \\
=n \times 2^{\frac{n-1}{2}}\left(\cos \frac{\pi}{4}+\mathrm{i} \sin \frac{\pi}{4}\right)^{n-1} \\
=n \times 2^{\frac{n-1}{2}}\left[\cos \frac{(n-1) \pi}{4}+\mathrm{i} \sin \frac{(n-1) \pi}{4}\right] .
\end{array}
$$

By comparing the real and imaginary parts, we get
$$
\begin{array}{l}
\text { (1) } \mathrm{C}_{n}^{1}-3 \mathrm{C}_{n}^{3}+5 \mathrm{C}_{n}^{5}-7 \mathrm{C}_{n}^{7}+\cdots \\
=n \times 2^{\frac{n-1}{2}} \cos \frac{(n-1) \pi}{4} \\
\text { (2) } 2 \mathrm{C}_{n}^{2}-4 \mathrm{C}_{n}^{4}+6 \mathrm{C}_{n}^{6}-8 \mathrm{C}_{n}^{8}+\cdots \\
=n \times 2^{\frac{n-1}{2}} \sin \frac{(n-1) \pi}{4}
\end{array}
$$