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3.Trigonometrical Ratios, Functions and Identities
medium
If $\tan x + \tan \left( {\frac{\pi }{3} + x} \right) + \tan \left( {\frac{{2\pi }}{3} + x} \right) = 3,$ then
A
$\tan x = 1$
B
$\tan 2x = 1$
C
$\tan 3x = 1$
D
None of these
Solution
(c) $\tan x + \tan \,\left( {\frac{\pi }{3} + x} \right) + \tan \,\left( {\frac{{2\pi }}{3} + x} \right)$
$ = \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
$\Rightarrow$ $3\tan 3x = 3$==> $\tan 3x = 1.$
Standard 11
Mathematics
