Question

Solve $\tan \theta +\tan (\theta +\pi /3)+\tan (\theta +2\pi /3)=3$

Answer 1

The given equation can be written as
$\tan x+\frac{\tan x+\tan \frac{\pi }{3}}{1-\tan x\tan \frac{\pi }{3}}+\frac{\tan x+\tan \frac{2\pi }{3}}{1-\tan x\tan \frac{2\pi }{3}}=3$
$\Rightarrow \tan x+\frac{\tan x+\sqrt{3}}{1-\sqrt{3}\tan x}+\frac{\tan x-\sqrt{3}}{1+\sqrt{3}\tan x}=3$
$\Rightarrow \tan x+\frac{8\tan x}{1-3{\tan }^{2}x}=3$
$\Rightarrow \frac{\tan x(1-3{\tan }^{2}x)+8\tan x}{1-3{\tan }^{2}x}=3$
$\Rightarrow \frac{3(3\tan x-{\tan }^{3}x)}{1-3{\tan }^{2}x}=3$
$\Rightarrow 3\tan 3x=3\Rightarrow \tan 3x=1$
$\Rightarrow 3\tan 3x=3\Rightarrow \tan 3x=1\Rightarrow 3x=n\pi +\frac{\pi }{4}\Rightarrow x=\frac{n\pi }{3}+\frac{\pi }{12}$