Find the general solution of each of the following equations:

To Find: General solution.

Given: $\cos x-\sin x=1 \Longrightarrow \cos \left(x+\frac{\pi}{4}\right)=\frac{-1}{\sqrt{2}}=\cos \frac{3 \pi}{4}$

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

So $\sin x=0$ or $\cos x=0$

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

$\Rightarrow x+\frac{\pi}{4}=2 k \pi \pm \frac{3 \pi}{4} \Rightarrow x=2 k \pi \pm \frac{3 \pi}{4}-\frac{\pi}{4} \Rightarrow x=2 k \pi+\frac{3 \pi}{4}-\frac{\pi}{4}$ or $\Rightarrow x=2 k \pi-\frac{3 \pi}{4}-$

$\frac{\pi}{4}$

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

So general solution is $x=2 n \pi+\frac{\pi}{2}$ or $x=(2 n-1) \pi$ where $n \in I$