Trigonometry 7th Edition

$$x=\frac{\pi}{5}+ \frac{ 2k\pi}{5}$$
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 .