$\begin{aligned} & \cos A+2 \cos B+\cos C=2 \\\\ & \cos A+\cos C=2(1-\cos B) \\\\ & 2 \cos \frac{A+C}{2} \cos \left(\frac{A-C}{2}\right)=2 \times 2 \sin ^2 \frac{B}{2} \\\\ & \cos \frac{A-C}{2}=2 \sin \frac{B}{2} \\\\ & 2 \cos \frac{B}{2} \cos \frac{A-C}{2}=4 \sin \frac{B}{2} \cos \frac{B}{2} \\\\ & 2 \sin \left(\frac{A+C}{2}\right) \cos \left(\frac{A-C}{2}\right)=2 \sin B \\\\ & \sin A+\sin C=2 \sin B \\\\ & a+c=2 b \quad(\because a=3, c=7) \\\\ & \Rightarrow b=5 \quad \end{aligned}$ $\begin{aligned} \cos A & -\cos C=\frac{b^2+c^2-a^2}{2 b c}-\frac{a^2+b^2-c^2}{2 \mathrm{ab}} \\\\ & =\frac{25+49-9}{70}-\frac{9+25-49}{30} \\\\ & =\frac{65}{70}+\frac{1}{2}=\frac{20}{14}=\frac{10}{7} \end{aligned}$