Prove that:

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

Question

Mathematics

Vicente Santos

22 June, 05:49

Prove that:

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

+2

Answers (1)

Know the Answer?

Rory Pacheco · Answer 1

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

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

Prove that:

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