Question

The number of values of $x$ in $[0,2\pi ]$ which satisfy $\tan x+\tan 4x+\tan 7x=\tan x\tan 4x\tan 7x$ is

Answer 1

$\tan x+\tan 4x+\tan 7x=\tan x\tan 4x\tan 7x$
$(\tan A+\tan B+\tan C=\tan A\tan B\tan C)\Rightarrow \tan (A+B+C)=0$
$\therefore \tan (x+4x+7x)=0$
$\tan \left(12x\right)=0$
$\Rightarrow 12x=n\pi ,n\in Z$
$\Rightarrow x=\frac{n\pi }{12},n\in Z$
Where $x\in [0,2\pi ]$
$\therefore x=0,\frac{\pi }{12},\frac{\pi }{6},\frac{\pi }{4},...,\frac{23\pi }{12},2\pi$
But $\tan x$ is not defined when $x=\frac{\pi }{2},\frac{3\pi }{2}$
So, number of values of $x$ is $23$