How do I evaluate cos(pi/5) without using a calculator?

##Cos (pi/5) =
(sqrt (5)+1)/4##.
Let ##a = cos(pi/5)## ##b = cos(2*pi/5)##. Thus ##cos(4*pi/5) = -a##. From the double angle formulas:
##b = 2a^2-1##
##-a = 2b^2-1##
Subtracting
##a+b = 2(a^2-b^2)
= 2(a+b)(a-b)##
##a+b## is not zero as both terms are positive so ##a-b## must be ##1/2##. Then
##a-1/2 = 2a^2-1##
##4a^2-2a-1 = 0##
and the only positive root is
##a = cos (pi/5) = (sqrt(5)+1)/4##.
And ##b = cos (2*pi/5)
= a-1/2 = (sqrt(5)-1)/4##.