Question

Find the general solution of $\cos x+\sin x=1$

Answer 1

We have,

$\cos x+\sin x=1$

$\Rightarrow \frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x=\frac{1}{\sqrt{2}}$ ....[Dividing both sides by $\sqrt{2}$]

$\Rightarrow \cos \frac{\pi }{4}\cos x+\sin \frac{\pi }{4}\sin x=\frac{1}{\sqrt{2}}$

$\Rightarrow \cos (x-\frac{\pi }{4})=\cos \frac{\pi }{4}$

$\therefore x-\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4}$, where $n\in I$

$\Rightarrow x=2n\pi \pm \frac{\pi }{4}+\frac{\pi }{4}$, where $n\in I$

$\Rightarrow x=2n\pi +\frac{\pi }{4}+\frac{\pi }{4},2n\pi -\frac{\pi }{4}+\frac{\pi }{4}$, where $n\in I$

$\Rightarrow x=2n\pi +\frac{\pi }{2},2n\pi$, where $n\in I$

$\Rightarrow x=(4n+1)\frac{\pi }{2},2n\pi$, where $n\in I$

This is the required general solution.