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22 October, 10:50

Prove that:

(tan20°) ^2 + (tan40°^) 2 + (tan80°) ^2=33

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  1. 22 October, 10:57
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    Step-by-step explanation:

    We have tanπ3=-tan2π3=tan4π3=3-√

    Since tan (3x) = 3tanx-tan3x1-3tan2x

    Thus tanπ/9,-tan2π/9, tan4π/9are solutions to the polynomial 3-√=3x-x31-3x2

    On simplication, x3-33-√x2-3x+3-√=0. (1)

    Since (tanπ9-tan2π9+tan4π9) 2

    =tan2π9+tan22π9+tan24π9

    + 2 (-tanπ9tan2π9+tanπ8tan4π9-tan2π9tan4π9)

    By sum and product pairs of roots in (1) above

    (-33-√) 2=tan2π9+tan22π9+tan24π9+2*-3

    ∴tan2π9+tan22π9+tan24π9=33
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