... or whistleralley.com/polyhedra/pentagon.htm for a nice account of the geometry of a regular pentagon and its connection with the golden ratio.
Working out in degrees, we have
Let x =72 degrees, multiply both sides by 5;
5x = 360
2x + 3x = 360
2x = 360-3x
Then, taking cos of both side
cos 2x = cos(360 -3x)
cos 2x = cos 3x
Using an identity for double angle and triple angle, equating them we have
2cos^2 x -1= 4cos^3 x - 3 cos x
4cos^3 x - 2cos^2 x - 3 cos x +1=0
4cos^3 x - 4cos^2 x + 2 cos^2 x - 2 cos x - cos x +1=0
4cos^2x(cos x -1) +2cos x( cos x -1) -1( cos x - 1) =0
(cos x - 1) (4cos^x +2 cos x - 1) =0
Either cos x =0 means x=90 degree
Solve (4cos^x +2 cos x - 1) =0
cos x = [-2 +(plus or minus){sq rt (4+16)] /8
cos x = [-2 +(plus or minus){sq rt (20)] /8
cos x = [-2 +2(plus or minus){sq rt (5)] /8
cos x = [-1 +(plus or minus){sq rt (5)] /4
Neglecting NEGATIVE values of cos x, because x lies in first quadrant
Therefore cos 72 degrees = (sqrt(5) -1)/4
cos 72 = 0.30901699437494742410229341718 . . . .