Question

The number of solution(s) of the equation $\tan x\tan 4x=1$ for $0<x<\pi$ is

• A. $1$
• B. $2$
• C. $4$
• D. $5$

Answer 1

The correct option is C $4$

$\tan x\tan 4x=1$
$\Rightarrow \cos 4x\cos x-\sin 4x\sin x=0$
$\Rightarrow \cos 5x=0$
$\Rightarrow 5x=(2n\pi +)\frac{\pi }{2},n\in Z$
$\Rightarrow x=\frac{(2n+1)\pi }{10};0<x<\pi$
$=\frac{\pi }{10},\frac{3\pi }{10},\frac{\pi }{2},\frac{7\pi }{10},\frac{9\pi }{10}$
But for $\frac{\pi }{2},\tan x$ is not defined
Thus, there are $4$ solutions only.