The solution of $\left|\cos \mathrm{x}\right| =\mathrm{cosx}-2\sin \mathrm{x} \mathrm{is}$

 $$\begin{gathered} \left|\cos \mathrm{x}\right| =\mathrm{cosx}-2\sin \mathrm{x} \\ We have, \mid \cos x\mid =\cos x-2\sin x \\ Case 1) if \cos x\ge 0 \\ \cos x=\cos x-2\sin x \\ \Rightarrow \sin x=0\Rightarrow x=n\pi \\ \therefore x=2n\pi \end{gathered}$$
$$\begin{gathered} Case 2) If \cos x<0 \\ -\cos x=\cos x-2\sin x \\ \Rightarrow \cos x-\sin x=0 \\ \Rightarrow \tan x=1 \\ \Rightarrow x=n\pi + \frac{\pi }{4} \end{gathered}$$