Example 4. Convert $\cos \alpha+\cos 3 \alpha+\cos 5 \alpha$ $+\cos 7 \alpha$ into a product.
Let $z=\cos \alpha+i \sin \alpha$. Then
$$
\begin{aligned}
\text { Original expression }= & \frac{1}{2}\left(z+z^{-1}+z^{3}+z^{-3}+z^{8}\right. \\
& \left.+z^{-5}+z^{7}+z^{-7}\right) \\
= & \frac{1}{2}\left(z+z^{-1}\right)\left(z^{2}+z^{-2}\right) \\
& \cdot\left(z^{4}+z^{-4}\right) \\
= & \frac{1}{2} \cdot 2 \cos \alpha \cdot 2 \cos 2 \alpha \cdot 2 \cos 4 \alpha \\
= & 4 \cos \alpha \cos 2 \alpha \cos 4 \alpha .
\end{aligned}
$$