Let #theta=pi/5#, then #5theta=pi#
and #3theta=pi-2theta#. Note #theta) is an acute angle.
Hence #sin3theta=sin(pi-2theta)# but as #sin(pi-A)=sinA#
This can be written as
#sin3theta=sin2theta# expanding them
or #3sintheta-4sin^3theta=2sinthetacostheta#
as #theta=pi/5# we have #sintheta!=0# and dividing by it we get
#3-4sin^2theta=2costheta# or
#3-4(1-cos^2theta)=2costheta#
or #4cos^2theta-2costheta-1=0#
and using quadratic formula #costheta=(2+-sqrt(2^2-4*4*(-1)))/(2*4)#
= #(2+-sqrt(20))/8=(1+-sqrt5)/4#.
But as #(1-sqrt5)/4# is negative and #costheta# cannot take this value, hence
#costheta=(1+sqrt5)/4# and
#sintheta=sqrt(1-((1+sqrt5)/4)^2)=sqrt(1-((1+5+2sqrt5)/16))#
= #sqrt((16-6-2sqrt5)/16)=sqrt(10-2sqrt5)/4#
As #theta=pi/5#
#sin(pi/5)=sqrt(10-2sqrt5)/4#