8.263. $\cos 3 x+\cos \frac{5 x}{2}=2$.

8.263. $\cos 3 x+\cos \frac{5 x}{2}=2$.
## Solution.

The equation is equivalent to a system of two equations

$$
\left\{\begin{array} { l } 
{ \operatorname { cos } 3 x \neq 1 , } \\
{ \operatorname { cos } \frac { 5 x } { 2 } = 1 , }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ 3 x = 2 \pi k , k \in Z , } \\
{ \frac { 5 x } { 2 } = 2 \pi n , n \in Z , }
\end{array} \Leftrightarrow \left\{\begin{array}{ll}
x=\frac{2 \pi k}{3}, k \in Z, \\
x=\frac{4 \pi n}{5}, n \in Z .
\end{array} \Rightarrow \begin{array}{l}
k=6 m \\
n=5 m
\end{array}\right.\right.\right.
$$

$$
\frac{2 \pi k}{3}=\frac{4 \pi n}{5}, 5 \pi k=6 \pi n, 5 k=6 n
$$

Answer: $x=4 \pi m, m \in Z$.