Question

prove that cos π/5=√5+1/4​

Answer 1

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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#.