cos 3x – cos 5x = sin x

\[ \Leftrightarrow \]2sin 4x sin x = sin x

\[ \Leftrightarrow \] sin x (2sin 4x – 1) = 0

\[ \Leftrightarrow \left[ \begin{array}{l}\sin x = 0\\2\sin 4x - 1 = 0\end{array} \right.\]

\[ \Leftrightarrow \left[ \begin{array}{l}\sin x = 0\\\sin 4x = \frac{1}{2}\end{array} \right.\]

\[ \Leftrightarrow \left[ \begin{array}{l}x = k\pi \\4x = \frac{\pi }{6} + k2\pi \\4x = \frac{{5\pi }}{6} + k2\pi \end{array} \right.\]

\[ \Leftrightarrow \left[ \begin{array}{l}x = k\pi \\x = \frac{\pi } + k\frac{\pi }{2}\\x = \frac{{5\pi }} + k\frac{\pi }{2}\end{array} \right.\,\,\,(\,k \in \mathbb{Z})\]

Vậy phương trình đã cho có các nghiệm là \[x = k\pi \]; \[x = \frac{\pi } + k\frac{\pi }{2}\]; \[x = \frac{{5\pi }} + k\frac{\pi }{2}\]\[(\,k \in \mathbb{Z})\]