4. Solve the equation $8 \cos ^{5} x-5 \cos x-2 \cos 3 x=1$.

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# Solution:

We need to figure out how to express $\cos ^{5} x$ in terms of $\cos 5 x$. Then the original equation will be reduced to $\cos 5 x + \cos 3 x = 2$. The last equality is only possible if the condition $\left\{\begin{array}{l}\cos 5 x=1 \\ \cos 3 x=1\end{array} \Leftrightarrow x=2 \pi n, n \in Z\right.$ is satisfied.

Answer: $x=2 \pi n, n \in Z$.