\begin{array}{l}\text { 1. Let } \cos A+\cos B+\cos C \\ =\sin A+\sin B+\sin C=0 \text {. } \\ \text { Then } \cos ^{4} A+\cos ^{4} B+\cos ^{4} C=\end{array}
```
Given $z_{1}=\cos A+\mathrm{i} \sin A$, etc. Then $\left|z_{j}\right|=1(j=1,2,3), z_{1}+z_{2}+z_{3}=0$.
It is easy to prove $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=0, z_{1}^{4}+z_{2}^{4}+z_{3}^{4}=0$.
\[
\begin{array}{l}
\text { Therefore, the original expression }=\frac{1}{4} \sum(1+\cos 2 A)^{2} \\
=\frac{1}{4} \sum\left(1+\frac{z_{1}^{2}+\overline{z_{1}^{2}}}{2}\right)^{2} \\
=\frac{3}{4}+\frac{1}{4} \sum\left(z_{1}^{2}+\overline{z_{1}^{2}}\right)+\frac{1}{16} \sum\left(z_{1}^{4}+\overline{z_{1}^{4}}+2\right) \\
=\frac{9}{8} .
\end{array}
\]
```