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

Options

A. 7
B. `35//3`
C. `21//5`
D. none of these

Answer

C

Explanation

Let `theta = (npi)/(7)` (so that `7theta = npi`), `n in Z`
`rArr 4theta + 3theta = npi`
`or 4theta = tan (npi - 3theta ) = -tan 3theta`
or ` (4tan theta - 4tan^(3)theta)/(1-6tan^(2) theta + tan^(4)theta) =- (3tan theta - tan^(3) theta)/(1-3tan^(2) theta )`
or ` " " ( 4z - 4z^(3))/(1-6z^(2)+ z^(4)) = - (3z-z^(3))/(1-3z^(2))" "` [where `tan theta =z` (say)]
or `(4-4z^(2)) (1-3z^(2))= -(3-z^(2)) (1-6z^(2)+z^(4))`
or ` z^(6) - 21 z^(4) = 35 z^(2) - 7 =0" "` ... (i)
This is a cubic equation in `z^(2)`, i.e., in `tan^(2)theta`.
The roots of this equation are therefore
`" " tan ^(2)pi//7, tan^(2) 2pi//7, and tan^(2) 3pi//7`. From Eq. (i),
`" "` Sum of the roots `= (-(-21))/(1) = 21`
`rArr tan^(2)""(pi)/(7) + tan^(2) ""(2pi)/(7) + tan^(2)"" (3pi)/(7) = 21" " `...(ii)
Putting `1//y` in place of z in Eq. (i), we get
`" " -7y^(6) + 35y^(4)- 21y^(2) +1 =0`
`or 7y^(6)- 35 y^(4)+ 21y^(2) -1=0" "`...(iii)
This is a cubic equation in `y^(2)`, i.e., in `cot^(2) theta`.
The roots of this equation are therefore `cot^(2)pi//7, cot^(2) 2pi//7, and cot^(2)3pi//7`.
Sum of the roots of Eq. (iii) = `(35)/(7) =5`
`rArr cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) +cot ^(2)""(3pi)/(7)=5" "` ...(iv)
By multiplying Eqs. (ii) and (iv), we get
`(tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2) ""(3pi)/(7))(cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7))`
`= 21xx5 = 105`