Question

If $n$ is any integer, then the general solution of the equation $\cos x-\sin x=\frac{1}{\sqrt{2}}$ is

• A. $x=2n\pi -\frac{\pi }{12}$ or $x=2n\pi +\frac{7\pi }{12}$
• B. $x=n\pi \pm \frac{\pi }{12}$
• C. $x=2n\pi +\frac{\pi }{12}$ or $x=2n\pi -\frac{7\pi }{12}$
• D. $x=n\pi +\frac{\pi }{12}$ or $x=n\pi -\frac{7\pi }{12}$

Answer 1

Answer: (C)

Given equation is $\cos x-\sin x=\frac{1}{\sqrt{2}}$
Dividing equation by $\sqrt{2}$,
$\frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x=\frac{1}{2}$
$\Rightarrow \cos \left(\frac{\pi }{4}+x\right)=\cos \frac{\pi }{3}$
$\Rightarrow \frac{\pi }{4}+x=2n\pi \pm \frac{\pi }{3};$
$x=2n\pi +\frac{\pi }{3}-\frac{\pi }{4}=2n\pi +\frac{\pi }{12}$
or $x=2n\pi -\frac{\pi }{3}-\frac{\pi }{4}=2n\pi -\frac{7\pi }{12}$.