how do we know the first rule?
We know that cos [2* (Pi/5)] = -cos [Pi - 2* (Pi/5)] =
= - cos [3* (Pi/2)]
Expanding both sides and putting cos(Pi/5) = x we get:
2x^2-1= -( 4x^3-3x) or
4x^2-2x-1 =0 gives: x= (2+sqrt20)/(2*4) or x= (2-sqrt20)/8 or x=-1,
But Pi/5 is an acute angle (equal t0 36 degree). So co 36 is a positive angle and x= (2+sqrt20)/8 or (1+sqrt5)/4 only valid.
So x = (1+sqrt5)/4 or
cos(Pi/5) = (1+sqrt5)/4