proof left as an exercise

[Albert Chan]

Using complex numbers, proof turns out very simple !

Let z = cis(20°), 1° = pi/180

cos(60°) = (z^3+1/z^3)/2 = 1/2      → (z^3+1/z^3) = 1

1 + 4*cos(20°)
= (z^3+1/z^3) + 2*(z+1/z)
= (z^2+1)*(z^4+z^2+1) / z^3
= (z^2+1)/(z^2-1) * (z^6-1)/z^3
= (2*cos(20°)) / (2i*sin(20°)) * (2i*sin(60°))
= 2*sin(60°) / tan(20°)

--> 2*cos(30°) / (1 + 4*sin(70°)) = tan(20°)