The value of `sin(pi/14)sin(3pi/14)sin(5pi/14)sin(7pi/14)sin(9pi/14)sin(11pi/14)sin(13pi/14)` is equal to___________

Answer

`1//64`

Explanation

`sin""(pi)/(14)sin""(5pi)/(14)sin""(7pi)/(14)sin""(9pi)/(14)sin""(11pi)/(14)sin""(13pi)/(14)`
`=sin""(pi)/(14)sin""(3pi)/(14)sin""(5pi)/(14)sin""(pi)/(2)sin""(pi-(5pi)/(14))`
`xxsin""(pi-(3pi)/(14))sin(x-(pi)/(14))`
`=sin^(2)""(pi)/(14)sin^(2)""(3pi)/(14)sin^(2)""(5pi)/(14)`
`=(sin ""(pi)/(14)sin ""(3pi)/(14)sin""(5pi)/(14))^(2)`
`=[cos((pi)/(2)-(pi)/(14))cos((pi)/(2)-(3pi)/(14))cos((pi)/(2)-(5pi)/(14))]^(2)`
`=[cos""(3pi)/(7)cos""(2pi)/(7)cos""(pi)/(7)]^(2)`
`[(1)/(2sin pi//7){2cos""(pi)/(7)sin""(pi)/(7)cos ""(2pi)/(7)cos""(3pi)/(7)}]^(2)`
`=[(1)/(2^(3)sinpi//7)(2sin""(4pi)/(7)cos(pi-(4pi)/(7))]^(2)`
`=((1)/(8sinpi//7)sin""(8pi)/(7))^(2)`
`=((sin(pi+pi//7)/(8sinpi//7)))^(2)`
`=((-sinpi//7)/(8sinpi//7))^(2)=((1)/(8))^(2)=(1)/(64)`