To find the sum `sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7)`, we follow the following method.
Put `7theta = 2npi`, where `n ` is any integer. Then
`" " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta`
This means that `sin theta` takes the values `0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7)`.
From Eq. (i), we now get
`" " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta `
or `4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3)`
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 (1-2 sin^(2) theta)^(2) = ( 4sin ^(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`
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))/(cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7))` is