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Question
The value of \[\tan x + \tan \left( \frac{\pi}{3} + x \right) + \tan \left( \frac{2\pi}{3} + x \right)\] is
Options
3 tan 3x
tan 3x
3 cot 3x
cot 3x
Solution
3 tan 3x
\[\frac{\pi}{3} = 60°, \frac{2\pi}{3} = 120° \]
\[\tan x + \tan\left( 60° + x \right) + \tan\left( 120° + x \right) = \tan x + \frac{\tan60° + \text{ tan } x}{1 - \tan60° \text{ tan } x} + \frac{\tan 120° + \tan x}{1 - \tan 120° \text{ tan } x}\]
\[ = \tan x + \frac{\sqrt{3} + \tan x}{1 - \sqrt{3}\tan x} + \frac{\left( - \sqrt{3} + \text{ tan } x \right)}{1 + \sqrt{3}\text{ tan } x} \]
\[ = \frac{\tan x\left( 1 - 3 \tan^2 x \right) + \left( \sqrt{3} + \text{ tan } x \right)\left( 1 + \sqrt{3}\text{ tan } x \right) + \left( - \sqrt{3} + \text{ tan } x \right)\left( 1 - \sqrt{3}\text{ tan } x \right)}{1 - 3 \tan^2 x}\]
\[ = \frac{\tan x - 3 \tan^3 x + \sqrt{3} + 3\text{ tan } x + \text{ tan } x + \sqrt{3} \tan^2 x + \text{ tan } x - \sqrt{3} \tan^2 x - \sqrt{3} + 3\text{ tan } x}{1 - 3 \tan^2 x}\]
\[ = \frac{9\text{ tan } x - 3 \tan^3 x}{1 - 3 \tan^2 x}\]
\[ = \frac{3\left( 3\text{ tan } x - \tan^3 x \right)}{1 - 3 \tan^2 x}\]
\[ = 3\tan3x\]
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