Find the general solution of each of the following equations:

To Find: General solution.

Given: $\cos x+\sin x=1 \Rightarrow \cos \left(x-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}=\cos \frac{\pi}{4}$

[divide $\sqrt{2}$ on both sides and $\cos (x-y)=\cos x \cos y-\sin x \sin y$ ]

Formula used: $\cos \theta=\cos \alpha \Rightarrow \theta=2 k \pi \pm \alpha, k \in I$

$\Rightarrow x-\frac{\pi}{4}=2 k \pi \pm \frac{\pi}{4} \Rightarrow x=2 k \pi \pm \frac{\pi}{4}+\frac{\pi}{4} \Rightarrow x=2 k \pi+\frac{\pi}{4}+\frac{\pi}{4}$ or $\Rightarrow x=2 k \pi-\frac{\pi}{4}+\frac{\pi}{4}$

$\Rightarrow x=2 k \pi+\frac{\pi}{2}$ or $x=2 k \pi$

So general solution is $x=2 n \pi+\frac{\pi}{2}$ or $x=2 n \pi$ where $n \in$ ।