Ta có:

\(\cos 5x + \cos x = \sin 2x - \sin 4x\)\( \Leftrightarrow 2\cos 3x.\cos 2x = - 2\cos 3x\sin x\)\( \Leftrightarrow 2\cos 3x\left( {\cos 2x + \sin x} \right) = 0\)\( \Leftrightarrow 2\cos 3x\left( { - 2{{\sin }^2}x + \sin x + 1} \right) = 0\)\( \Leftrightarrow 2\cos 3x\left( {1 - \sin x} \right)\left( {2\sin x + 1} \right) = 0\)

\(\Leftrightarrow \left[ \begin{array}{l}\cos 3x = 0\\\sin x = 1\\\sin x = - \dfrac{1}{2}\end{array} \right. \)\(\Leftrightarrow \left[ \begin{array}{l}3x = \dfrac{\pi }{2} + k\pi \\x = \dfrac{\pi }{2} + k2\pi \\x = - \dfrac{\pi }{6} + k2\pi \\x = \dfrac{{7\pi }}{6} + k2\pi \end{array} \right. \)\(\Leftrightarrow \left[ \begin{array}{l}x = \dfrac{\pi }{6} + k\dfrac{\pi }{3}\\x = \dfrac{\pi }{2} + k2\pi \\x = - \dfrac{\pi }{6} + k2\pi \\x = \dfrac{{7\pi }}{6} + k2\pi \end{array} \right.\quad \left( {k \in \mathbb{Z}} \right)\)

Các nghiệm thuộc đoạn \(\left[ { - \pi ;\pi } \right]\) là \(\left\{ { - \dfrac{{5\pi }}{6}; - \dfrac{\pi }{2}; - \dfrac{\pi }{6};\dfrac{\pi }{6};\dfrac{\pi }{2};\dfrac{{5\pi }}{6}} \right\}\)

Tổng các nghiệm bằng: 0