$\cos^3x+\sin^3x=\sin x-\cos x$
$\to \cos x(\cos^2x+1)=\sin x(1-\sin^2x)$
$\to \cos x(\cos^2x+1)=\sin x.\cos^2x$
$\to \cos^2x(\cos x-\sin x)+\cos x=0$
$\to \cos x[\cos x(\cos x-\sin x)+1]=0$
$\to \cos x(\cos^2x-\sin x\cos x+1)=0$
$\to \cos x\left( \dfrac{1}{2}+\dfrac{1}{2}\cos2x-\dfrac{1}{2}\sin2x+1\right)=0$
$\to \dfrac{1}{2}\cos x(\cos2x-\sin2x+3)=0$
$\to \cos x.\left[ \sqrt2\cos(2x+\dfrac{\pi}{4})+3\right)=0$
$\to \left[\begin{matrix} \cos x=0\\ \cos\left(2x+\dfrac{\pi}{4}\right)=\dfrac{-3}{\sqrt2}<-1(L)\end{matrix}\right.$
$\to x=\dfrac{\pi}{2}+k\pi\quad(k\in\mathbb{Z})$
