$\begin{array}{ll}\text { Given, } & \sin x-\cos x=\sqrt{2} \\ \Rightarrow & \frac{1}{\sqrt{2}} \sin x-\frac{1}{\sqrt{2}} \cos x=1 \\ \Rightarrow & \sin x \cdot \sin \frac{\pi}{4}-\cos x \cdot \cos \frac{\pi}{4}=1 \\ \Rightarrow & -\cos \left(x+\frac{\pi}{4}\right)=+1 \\ \Rightarrow & \cos \left(x+\frac{\pi}{4}\right)=\cos \pi \\ \Rightarrow & x+\frac{\pi}{4}=2 n \pi+\pi \\ \therefore & x=2 n \pi+\frac{3 \pi}{4}\end{array}$