For example, $3 z \in \mathbf{C}, z^{7}=1(z \neq 1), z$ has an argument of $\alpha$, find the value of $\cos \alpha+\cos 2 \alpha+\cos 4 \alpha$.
Given $z \cdot \bar{z}=1$, it is easy to derive $z^{6}=\bar{z}, z^{5}=\bar{z}^{2}, z^{4}=\bar{z}^{3}$ (Property 2), and $1+z+z^{2}+\cdots+z^{6}=0$. Therefore, $1+z+z^{2}+z^{4}+\overline{z^{4}+z^{2}+z}=0, z+z^{2}+z^{4}+\overline{z+z^{2}+z^{4}}=-1$, which means $\cos \alpha+\cos 2 \alpha+$ $\cos 4 \alpha=-\frac{1}{2}$.