1. $-\frac{11}{14}$ or $-\frac{13}{14}$.
\[
\left.\begin{array}{l}
A, B, C \text { form an arithmetic sequence } \\
A+B+C=180^{\circ}
\end{array}\right\} \Rightarrow B=\frac{1}{3}(A+B+C)=60^{\circ}, \frac{8}{\sin A}=\frac{7}{\sin 60^{\circ}} \Rightarrow \sin A=\frac{8}{7} \sin 60^{\circ}=
\]
\[
\begin{array}{l}
\frac{4 \sqrt{3}}{7}, \cos A= \pm \sqrt{1-\sin ^{2} A}= \pm \frac{1}{7}, \cos C=\cos \left[180^{\circ}-(A+B)\right]=\cos (A+B)=\cos (A+ \\
60^{\circ}), \cos A \cos 60^{\circ}-\sin A \sin 60^{\circ}=-\frac{11}{14} \text { or }-\frac{13}{14} .
\end{array}
\]
1. In $\triangle A B C$, the internal angles $A, B, C$ form an arithmetic sequence, the side lengths $a=8, b=7, \cos C=$