Ta có: \(\cos 3x-\cos 5x=\sin x\)
\(\Leftrightarrow \sin x+\cos 5x-\cos 3x=0\)
\(\Leftrightarrow \sin x-2\sin\dfrac{5x+3x}{2}\sin\dfrac{5x-3x}{2}=0\)
\(\Leftrightarrow \sin x-2\sin 4x\sin x=0\)
\(\Leftrightarrow \sin x(1-2\sin 4x)=0\)
\(\Leftrightarrow \left[ \begin{array}{l} \sin x = 0\\\sin 4x= \dfrac{1}{2}\end{array} \right. \)
\(\Leftrightarrow \left[ \begin{array}{l} x = k\pi,k\in\mathbb{Z}\\4x= \dfrac{\pi}{6}+k2\pi,k\in\mathbb{Z}\\4x=\pi-\dfrac{\pi}{6}+k2\pi,k\in\mathbb{Z}\end{array} \right. \)
\(\Leftrightarrow \left[ \begin{array}{l} x = k\pi,k\in\mathbb{Z}\\x= \dfrac{\pi}{24}+k\dfrac{\pi}{2},k\in\mathbb{Z}\\x=\dfrac{5\pi}{24}+k\dfrac{\pi}{2},k\in\mathbb{Z}\end{array} \right. \)
