Question

The number of solutions of the equation $|\cos x–\sin x|=2\cos x,xϵ[0,2\pi ]$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is B

2

Three cases can be considered:

(i) $\cos x=\sin x=0$ which is impossible

(ii) $\cos x>\sin x,0\le x<\frac{\pi }{4}\text{and}\frac{5\pi }{4}<x\le 2\pi$

$\Rightarrow 2\cos x=\cos x-\sin x$

$\Rightarrow \tan x=-1\Rightarrow x=\frac{7\pi }{4}$

(iii) $\cos x<\sin x\Rightarrow \frac{\pi }{4}<x<\frac{5\pi }{4}$

then $x={\tan }^{-1}3$

So, there are 2 solutions.