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Lets take a = 2pi/7
7a =2pi sin4a = sin(2pi-3a) sin4a = -sin3a 2sin2a.cos2a = 4sin3(a) -3sina 4sin a.cos a (1-2sin2(a)) = sin a(4sin2(a) - 3) 4cos a(1-2sin2(a)) = 4sin2(a) -3 On squaring both sides
16(1-sin2(a)) [1-2sin2(a))]^2 = (4sin2(a) -3)2 64sin6(a) - 112sin4(a) + 56sin2(a) -7 =0 it is cubic in sin2(a) its roots are sin2(2pi/7) ,sin4(pi/7) ,sin2(8pi/7) sum of roots =7/4 sin2pi/7*sin4pi/7 +sin4pi/7*sin8pi/7 +sin8pi/7*sin2pi/7 = 0 we can simply prove it by using 2sin a.sin b= cos(a-b) - cos(a+b) & cos(2pi-theta) = cos theta (sin2pi/7+sin4pi/7 +sin8pi/7)2=7/4 sin2pi/7+sin4pi/7 +sin8pi/7=[(7)1/2]/2
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