$\sin ^3x+\cos ^3x=\sin x-\cos x$$\Leftrightarrow \sin x(\sin ^2x-1)+\cos x(\cos ^2x+1)=0$
$\Leftrightarrow -\sin x.\cos ^2x+\cos x(2\cos ^2x+\sin ^2x)=0$
$\Leftrightarrow\cos x(2\cos ^2x-\sin x.\cos x+\sin ^2x)=0$
$\Leftrightarrow \cos x=0 $
$\Leftrightarrow ...................$
~~~~$Amen$~~~~