Question

The general solution of the equation $\frac{\tan x}{\tan 2x}+\frac{\tan 2x}{\tan x}+2=0$ is

• A. $n\pi$
• B. $\frac{n\pi }{3}$
• C. $(3n\pm 1)\frac{\pi }{3}$
• D. $(2n-1)\frac{\pi }{3};n\in Z$

Answer 1

Answer: (C)

Put $\frac{\tan x}{\tan 2x}=y$
$\therefore y+\frac{1}{y}+2=0\Rightarrow y^2+2y+1=0$
$\Rightarrow (y+1{)}^2=0\Rightarrow y=-1$
$\therefore \frac{\tan x}{\tan 2x}=-1\Rightarrow \tan 2x=-\tan x$
$\Rightarrow \frac{2\tan x}{1-{\tan }^{2}x}=-\tan x$
$\Rightarrow 2=-1+{\tan }^{2}x[\because \tan x\ne 0]$
$\Rightarrow 3={\tan }^{2}x$
$\therefore \tan x=\pm \sqrt{3}\Rightarrow x=\pm \frac{\pi }{3}$
$\therefore$ general solution is $x=n\pi \pm \frac{\pi }{3}$
$i,e.,x=(3n\pm 1)\frac{\pi }{3}$