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If 1, a1, a2,....,a(n-1) are the nth roo...

If `1, a_1, a_2,....,a_(n-1)` are the nth roots of unity then prove that `(1-a_1)(1-a_2)(1-a_3)=(1-a_(n-1))=n`

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Since `1,a_1,a_2…a_(n+1)` are nth roots of unity `rArr (x^n-1)=(x-1)(x-a_1)(x-a_2)……(x-a_(n-1))`
`rArr (x^n-1)/(x-1)=(x-a_1)(x-a_2)….(x-a_(n-1))`
`rArr x^(n-1)+n^(n-2)+…..+x^2+x+1`
`rArr x^(n-1)+x^(n-2)+…….+x^2+x+1`
`=(x-a_1)(x-a_2)......(x-a_(n-1))`
`[ because (x^n-1)/(x-1)=x^(n-2)+....+x+1]`
On putting x=1 we get 1+1+...n times
`=(1-a_1)(1-a_2).....(1-a_(n-1))`
`rArr (1-a_1)(1-a_2)....(1-a_(n-1))=n`
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