A) \[\tan x=1\]
B) \[\tan 2x=1\]
C) \[\tan 3x=1\]
D) None of these
Correct Answer: C
Solution :
\[\tan x+\tan \,\left( \frac{\pi }{3}+x \right)+\tan \,\left( \frac{2\pi }{3}+x \right)\] \[=\tan x+\frac{\tan x+\sqrt{3}}{1-\sqrt{3}\,\tan x}+\frac{\tan x-\sqrt{3}}{1+\sqrt{3}\,\tan x}\] \[=\tan x+\frac{8\tan x}{1-3{{\tan }^{2}}x}=\frac{3\,(3\tan x-{{\tan }^{3}}x)}{1-3{{\tan }^{2}}x}=3\tan 3x\] Therefore, the given equation is \[3\tan 3x=3\] Þ \[\tan 3x=1.\]
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