[tex]\sin(x)-\cos(x)=\sqrt2\\\\
(\sin(x)-\cos(x))^2=(\sqrt2)^2\\\\
\sin^2(x)-2\sin(x)\cos(x)+\cos^2(x)=2\\\\
\underbrace{\sin^2(x)+\cos^2(x)}_{=1}-\underbrace{2\sin(x)\cos(x)}_{=\sin(2x)}=2\\\\
1-\sin(2x)=2\\\\
\sin(2x)=-1\Longrightarrow 2x=\dfrac{3\pi}{2}+2k\pi,~k\in\mathbb{Z}\\\\\Longrightarrow \boxed{x=\dfrac{3\pi}{4}+k\pi,~k\in\mathbb{Z}}[/tex]
