Find the general solution of each of the following equations:
cos x + sin x = 1
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 $n \in$ ।
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