Question

Solve:$|\cos x|=\cos x-2\sin x$

• A. $x=(2n+1)\pi +\frac{\pi }{4}$
• B. $x=(2n+1)\pi +\frac{\pi }{2}$
• C. $x=2n\pi +\frac{\pi }{4}$
• D. $x=2n\pi +\frac{\pi }{2}$

Answer 1

The correct option is A $x=(2n+1)\pi +\frac{\pi }{4}$

We have, $\left|\cos x\right|=\cos x-2\sin x$

Case 1)

if $\cos x\ge 0$

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

$\Rightarrow \sin x=0\Rightarrow x=n\pi$

$\therefore x=2n\pi$
Case 2)

If $\cos x<0$

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

$\Rightarrow \cos x-\sin x=1$

$\Rightarrow \tan x=1\Rightarrow x=n\pi +\frac{\pi }{4}$

$\therefore x=(2n+1)\pi +\frac{\pi }{4}$