Question

The general solution of $\cos 2 x-2 \tan x+2=0$ is

• A. $(2 n+1) \frac{\pi}{3}, n \in Z$
• B. $(n+1) \frac{\pi}{3}, n \in Z$
• C. $n \pi+\frac{\pi}{3}, n \in Z$
• D. $n \pi+\frac{\pi}{4}, n \in Z$

Answer 1

Answer: (D)

We have,
$\cos 2 x-2 \tan x+2=0$
$\Rightarrow \, \frac{1-\tan ^{2} x}{1+\tan ^{2} x}-2 \tan x+2=0$
$\Rightarrow \, \left(1-\tan ^{2} x\right)-2(\tan x-1)\left(1+\tan ^{2} x\right)=0$
$\Rightarrow \, (1-\tan x)\left(1+\tan x+2+2 \tan ^{2} x\right)=0$
$\Rightarrow \, (1-\tan x)\left(2 \tan ^{2} x+\tan x+3\right)=0$
$\Rightarrow \, \tan x=1 \Rightarrow \tan x=\tan \frac{\pi}{4}$
$\Rightarrow x=n \pi+\frac{\pi}{4}$