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Q.
If $n$ is any integer, then the general solution of the
equation $\cos x-\sin x=\frac{1}{\sqrt{2}}$ is
Trigonometric Functions
Solution:
Given equation is $\cos x-\sin x=\frac{1}{\sqrt{2}}$
Dividing equation by $\sqrt{2}$,
$\frac{1}{\sqrt{2}} \cos x-\frac{1}{\sqrt{2}} \sin x=\frac{1}{2} $
$\Rightarrow \cos \left(\frac{\pi}{4}+x\right)=\cos \frac{\pi}{3}$
$\Rightarrow \quad \frac{\pi}{4}+x=2 n \pi \pm \frac{\pi}{3} ;$
$ x=2 n \pi+\frac{\pi}{3}-\frac{\pi}{4}=2 n \pi+\frac{\pi}{12}$
or $\quad x=2 n \pi-\frac{\pi}{3}-\frac{\pi}{4}=2 n \pi-\frac{7 \pi}{12}$.