Let 1, `z_(1),z_(2),z_(3),…., z_(n-1)` be the nth roots of unity. Then prove that `(1-z_(1))(1 - z_(2)) …. (1-z_(n-1))= n`. Also,deduce that `sin .(pi)/(n) sin.(2pi)/(pi)sin.(3pi)/(n)...sin.((n-1)pi)/(n) = (pi)/(2^(n-1))`

Explanation

`z^(n)-1=(z-1)(z-z_(1))(z-z_(2))...(z-z_(n-1))`
`implies(z^(n)-1)/(z-1)=(z-z_(1))(z-z_(2))...(z-z_(n-1))`
`implies1+z+z^(2)+...z^(n-1)=(z-z_(1))(z-z_(2))...(z-z_(n-1))`
Putting `z=1`, we get
`(1-z_(1))(1-z_(2))...(1-z_(n-1))=1+1+...+1=n`
`implies|1-z_(1)||1-z_(2)|...|1-z_(n-1)|=n`
`=underset(r=1)overset(n-1)prod|1-z_(r)|=n`
`=underset(r=1)overset(n-1)prodsqrt((1-"cos"(2rpi)/(n))^(2)+("sin"(2rpi)/(n))^(2))=n`
`(becausez_(r)="cos"(2rpi)/(n)+i"sin"(2rpi)/(n))`
`impliesunderset(r=1)overset(n-1)prodsqrt((2"sin"^(2)(rpi)/(n))^(2)+(2"sin"(rpi)/(n)"cos"(rpi)/(n))^(2))=n`
`impliesunderset(r=1)overset(n-1)prod2"sin"(rpi)/(n)sqrt("sin"^(2)(rpi)/(n)+"cos"^(2)(rpi)/(n))=n`
`impliesunderset(r=1)overset(n-1)prod2"sin"(rpi)/(n)=n`
`impliesunderset(r=1)overset(n-1)prod"sin"((rpi)/(n))=(n)/(2^(n-1))`