Draw a 36^o-72^o-72^o triangle, with two long sides = 1, and the short side = x . 

Now, bisect one of the 72° angles forming two triangles:
a 36^o-36^o-108^o isosceles triangle whose short sides are both x , and a smaller 36^o-72^o-72^o isosceles triangle whose long  sides are x , and short side is 1−x  

Since the two 36^o-72^o-72^o triangles are proportional: <o:p></o:p>

x/1=(1-x)/x

solving      x²+x-1=0

and ignoring the negative answer

we get   x=(-1+sqrt(5))/2

a.k.a.     phi, the golden ratio<o:p></o:p>

so:

cos 36° =1/(-1+sqrt(5))=(1+phi)/2  =.809 approximately.