Question

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

• A. $n\pi$
• B. $\frac{n\pi}{3}$
• C. $(3n\pm1)\frac{\pi}{3}$
• D. $(2n-1)\frac{\pi}{3};n\in\,Z$

Answer 1

Answer: (C)

Put $\,\frac{\tan\,x}{\tan\,2x}=y$
$\therefore \,y+\frac{1}{y}+2=0 \Rightarrow \,y^2+2y+1=0$
$\Rightarrow \,(y+1)^2=0\,\Rightarrow \,y=-1$
$\therefore \:\: \frac{\tan x}{\tan 2x} = -1 \Rightarrow \tan 2x = - \tan x$
$\Rightarrow \,\frac{2\tan\,x}{1-\tan^2\,x}=-\tan\,x$
$\Rightarrow \, \, \, \, \, 2=-1+\tan^2\,x \, \, \, \, \, \, [\because\,\tan\,x\,\neq\,0]$
$\Rightarrow \, \, \, \, \, \, 3 = \tan^2 x$
$\therefore \: \, \, \tan\,x=\,\pm\sqrt{3}\Rightarrow \,x=\pm\,\frac{\pi}{3}$
$\therefore $ general solution is $x = n\pi\,\pm\frac{\pi}{3}$
$i,e.,\, \, \, \, \, \, x=(3n\pm1)\frac{\pi}{3}$