Question
Find the general solution of $\cos x+\sin x=1$.
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Answer
cos x + sinx = 1
$\cos x \frac{1}{\sqrt{2}}+\sin x \frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}$
$\cos x \frac{\cos \pi}{4}+\sin x \frac{\sin \pi}{4}=\frac{\cos \pi}{4}$
$\cos \left(x-\frac{\pi}{4}\right)=\cos \left(\frac{\pi}{4}\right)$
by $\cos y=\cos \alpha$
$y=2 n \pi \pm \alpha$
$x-\frac{\pi}{4}=2 n \pi \pm \frac{\pi}{4}$
$x-\frac{\pi}{4}=2 n \pi+\frac{\pi}{4}$ or $x-\frac{\pi}{4}=2 n \pi-\frac{\pi}{4}$
$x=2 n \pi+\frac{\pi}{2}$ or $x=2 n \pi$
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