11. $\cos ^{2} x+\cos ^{2} 2 x+\cos ^{2} 3 x=1$.

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11. $\cos ^{2} x+\cos ^{2} 2 x+\cos ^{2} 3 x=1$.
11. Using the formula $\cos ^{2} \frac{\alpha}{2}=\frac{1+\cos \alpha}{2}$ and replacing the sum $\cos 2 x+\cos 4 x$ with a product, we get $4 \cos 3 x \cos 2 x \cos x=0$.

Answer: $x_{1}=\frac{\pi}{6}+\frac{\pi}{3} k, \quad x_{2}=\frac{\pi}{2}+\pi k, \quad x_{3}=\frac{\pi}{4}+\frac{\pi}{2} k, \quad k=0 ; \quad \pm 1, \pm 2, \ldots$