Question

Prove that:
(tan20°)^2+(tan40°^)2+(tan80°)^2=33​

Answer 1

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