ĐK: $\cos x\ne 0\to x\ne \dfrac{\pi}{2}+k\pi$
$(\sin2x-\cos2x)\tan x+\dfrac{\sin3x}{\cos x}=\sin x+\cos x$
$\to \sin x(\sin2x-\cos2x)+\sin3x=\sin x\cos x+\cos^2x$
$\to \sin x\sin2x-\sin x\cos2x+\sin x\cos2x+\sin2x\cos x=\sin x\cos x+\cos^2x$
$\to \sin2x(\sin x+\cos x)=\cos x(\sin x+\cos x)$
$\to \sqrt2\sin\left(x+\dfrac{\pi}{4}\right).\cos x(2\sin x-1)=0$
$\to \left[ \begin{array}{l}\sin\left(x+\dfrac{\pi}{4}\right)=0 \\ \cos x=0 \\ \sin x=\dfrac{1}{2} \end{array} \right.$
$\to \left[ \begin{array}{l}x=\dfrac{-\pi}{4}+ k\pi\\x=\dfrac{-\pi}{2}+k\pi \\ x=\dfrac{\pi}{6}+k2\pi \\ x=\dfrac{5\pi}{6}+ k2\pi \end{array} \right.$
Đối chiếu ĐK, suy ra các họ nghiệm của PT là:
$\left[ \begin{array}{l}x=\dfrac{-\pi}{4}+k\pi \\ x=\dfrac{\pi}{6}+k2\pi \\ x=\dfrac{5\pi}{6}+k2\pi \end{array} \right.$
