Question

Find the general solution of the equation, $2+\tan x.\cot \frac{x}{2}+\cot x.\tan \frac{x}{2}=0$

• A. $x=2n\pi \pm \frac{2\pi }{3}$
• B. $x=2n\pi \pm \frac{\pi }{3}$
• C. $x=n\pi \pm \frac{2\pi }{3}$
• D. None of these

Answer 1

The correct option is A $x=2n\pi \pm \frac{2\pi }{3}$

Given, $2+\tan x.\cot \frac{x}{2}+\cot x.\tan \frac{x}{2}=0$

$\Rightarrow 2+\tan x.\cot \frac{x}{2}+\frac{1}{\tan x.\cot \frac{x}{2}}=0$ ...(1)

Let $\tan x.\cot \frac{x}{2}=y$,

then equation (1) becomes

$2+(y+\frac{1}{y})=0\Rightarrow y+\frac{1}{y}=-2$

$y^2+2y+1=0⟹(y+1{)}^2=0$

Solving, we get, $y=-1$

$\Rightarrow \tan x.\cot \frac{x}{2}=-1\Rightarrow \frac{\sin x}{\cos x}.\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}=-1$

$\Rightarrow \frac{2\sin \frac{x}{2}\cos \frac{x}{2}}{\cos x}.\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}=-1\Rightarrow \frac{2{\cos }^2\frac{x}{2}}{\cos x}=-1$

$\Rightarrow \frac{1+\cos x}{\cos x}=-1\Rightarrow \cos x=-\frac{1}{2}$

$\Rightarrow x=2n\pi \pm \frac{2\pi }{3}$