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Question

If tan x + tan x+π3+tan x+2π3=3, then prove that 3 tan x- tan3 x1-3 tan2 x=1.

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Solution

Given:tan x +tanx+π3+tanx+2π3 = 3tanx+tan x+tanπ31-tan x tan π3+tan x+tan2π31-tan x tan2π3=3tanx+tan x+31-3tan x +tan x-31+3tan x =3 tan120°=-3tan x(1-3tan2x)+tan x+3+3tan2x+3tan x+tan x-3-3tan2x+3tan x1-3tan2x=3 9tan x-3tan3x1-3tan2x=33tan x-tan3x1-3tan2x=1Hence proved.

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