show that `sin (2pi/7) + sin(4 pi/7) + sin (8 pi/7) = sqrt7/2`

Show that: sin((2pi)/7)+ sin ((4pi)/7) + sin( (8pi)/7) = sqrt7/2 .

Show that : sin pi/7 *sin 2pi/7* sin3pi/7 = sqrt7/8

sin "" (2pi)/(7) + sin "" (4pi)/(7) + sin "" (8pi)/(7) =

Show that : "sin" 2(pi)/(7)+ "sin"(4pi)/(7)-"sin" (6pi)/(7) =4 "sin" (pi)/(7) "sin" (2pi)/(7) "sin" (3pi)/(7)

Prove that: sin ((pi)/(7))+sin((2pi)/(7)) + sin((8pi)/(7)) + sin((9pi)/(7))=0

the value of sin(pi/7)+sin((2pi)/7)+sin((3pi)/7) is

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