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