To find the sum "sin"^(2) (2pi)/(7) + "sin"^(2) (4pi)/(7) + "sin"^(2) (8pi)/(7), we follow the following method. Put 7 theta = 2 n pi, where n is any integer. Then sin 4 theta = sin (2n pi - 3 theta) = - sin 3 theta""…(i) This means that sin theta takes the values 0. +- sin (2pi//7), +- sin (4pi//7), and +- sin (8pi//7). From Eq. (i), we now get 2 sin 2theta cos 2 theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta(1 - 2 sin^(2) theta) = (4 sin^(2) theta - 3) sin theta Rejecting the value sin theta = 0, we get 4 cos theta (1-2 sin^(2) theta) = 4 sin^(2) theta - 3 or 16 cos^(2) theta (1-2sin^(2)theta)^(2) = (4 sin^(2) theta - 3)^(2) or 16(1-sin^(2)theta)(1-4 sin^(2)theta + 4 sin^(4) theta) = 16 sin^(4) theta - 24 sin^(2) theta + 9 or 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta - 7 = 0, and this is cubic in sin^(2) theta with the roots sin^(2) ((2pi)/(7)),sin^(2)((4pi)/(7))and sin^(2) ((8pi)/(7)) The sum of these roots is "sin"^(2) (2pi)/(7) + "sin"^(2) (4pi)/(7) + "sin"^(2) (8pi)/(7) = (112)/(64) = (7)/(4). The value of "tan"^(2) (pi)/(7) "tan"^(2)(2pi)/(7) "tan"^(2) (3pi)/(7)