proof left as an exercise

[Albert Chan]

(06-08-2022 01:50 AM)Albert Chan Wrote:  Another proof, letting t = tan(10°)

tan(30°) = 1/√3 = (3t-t³) / (1-3t²)                  // triple-angle formula
⇒ t³ = √3*t² + 3t - 1/√3

We could simplify previous proof, by getting t² in terms of t

t²*t = √3*t² + 3t - 1/√3
t² = (3t - 1/√3) / (t - √3)

2*cos(30°) / (1 + 4*cos(20°))
= √3 / (1 + 4*(1-t²)/(1+t²))
= (1/√3 - t) / (1 + (1/√3)*t)
= (tan(30°) - tan(10°)) / (1 + tan(30°)*tan(10°))
= tan(20°)