Answer
$$x=\frac{\pi}{5}+ \frac{ 2k\pi}{5}$$
Work Step by Step
Given $$ \cos^3 5x = -1 $$
Then
$$ \cos 5x =\sqrt[3]{-1} =-1 $$
Then
$$ 5x=\pi+2k\pi \ \Rightarrow x=\frac{\pi}{5}+ \frac{ 2k\pi}{5}$$
where $k$ is any integer .
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