471. $\cos x + \cos 3x + \cos 5x = 0$.
Solution. Grouping the first and last terms and applying the cosine sum formula, we get

$$
(\cos x+\cos 5 x)+\cos 3 x=0 ; 2 \cos 3 x \cos x+\cos 3 x=0
$$

Therefore, $\cos 3 x(2 \cos x+1)=0$, from which $\cos 3 x=0$ or $2 \cos x+1=0$. Solving the equation $\cos 3 x=0$, we find $3 x= \pm \arccos 0+2 \pi k= \pm \frac{\pi}{2}+$ $+2 \pi k$, i.e., $x_{1}= \pm \frac{\pi}{6}+\frac{2 \pi k}{3}, k \in \mathbf{Z}$. Solving the equation $2 \cos x+1=0$, we have $\cos x=-\frac{1}{2}$, i.e., $x_{2}= \pm \arccos \left(-\frac{1}{2}\right)+2 \pi k= \pm \frac{2 \pi}{3}+2 \pi k, k \in \mathbf{Z}$.