The general solution of the trigonometrical equation $\sin x+\cos x=1\text{ for }n=0,\pm 1,\dots$ is given by

Solution:

We have

$\begin{array}{l}\sin x+\cos x=1\\ \Rightarrow \sqrt{2}\left(\sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4}\right)=1\\ \Rightarrow \sin \left(x+\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}\\ \Rightarrow \sin \left(x+\frac{\pi }{4}\right)=\sin \frac{\pi }{4}\\ \Rightarrow x+\frac{\pi }{4}=n\pi +(-1{)}^n\frac{\pi }{4},n\in Z\end{array}$

$\Rightarrow x=n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{4},n\in Z$