The Value of \( \tan x+\tan\left(\frac{\pi}{3}+x\right)+\tan\left(\frac{2\pi}{3}+x\right) \)
Question
Find the value of
\[ \tan x+\tan\left(\frac{\pi}{3}+x\right)+\tan\left(\frac{2\pi}{3}+x\right) \]
(a) \(3\tan3x\)
(b) \(\tan3x\)
(c) \(3\cot3x\)
(d) \(\cot3x\)
Solution
Let
\[ A=x,\quad B=x+\frac{\pi}{3},\quad C=x+\frac{2\pi}{3} \]
Then
\[ A+B+C=3x+\pi \]
Using the identity
\[ \tan(A+B+C) = \frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C} {1-\tan A\tan B-\tan B\tan C-\tan C\tan A} \]
For angles differing by \(60^\circ\),
\[ \tan A\tan B+\tan B\tan C+\tan C\tan A=3 \]
and
\[ \tan A\tan B\tan C=\tan3x \]
Hence,
\[ \tan A+\tan B+\tan C = 3\cot3x \]
Therefore,
\[ \tan x+\tan\left(\frac{\pi}{3}+x\right)+\tan\left(\frac{2\pi}{3}+x\right) = 3\cot3x \]
Final Answer
\[ \boxed{3\cot3x} \]
Hence, the correct option is (c) \(3\cot3x\).