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Question

If tanx+tan(x+π3)+tan(x+2π3)=3, then prove that 3tanxtan3x13tan2x=1

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Solution

We have,
tanx+tan(x+π3)+tan(x+2π3)=3
tanx+[tanx+tanπ31tanx×tanπ3]+[tanx+tan(2π3)1tanx×tan2π3]=3
tanx+[tanx+313tanx]+[tanx+tan(π2+π3)1tanxtan(π2+π3)]=3
tanx+tanx+313tanx+tanx31+3tanx=3
tanx+tanx+313tanx+tanx31+3tanx=3
tanx+(tanx+3)(1+3tanx)+(tanx3)(13tanx)(13tanx)(1+3tanx)=3
tanx+tanx+3tan2x+3+3tanx+tanx3tan2x3+3tanx1(3tanx)2=3
tanx+8tanx13tan2x=3tanx(13tan2x)+8tanx13tan2x=3
tanx3tan3x+8tanx13tan2x=39tanx3tan3x13tan2x=3
3(3tanxtan2x)13tan2x3tanxtan3x13tan2x=1
Hence proved.

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