If ω=cosπn+isinπn, then value of 1+ω+ω2+…+ωn-1 is

We know, 1+ω+ω2+…+ωn−1=1−ωn1−ω But ωn=cosnπn+isinnπn=cosπ+isinπ=−1 and 1−ω=2sin2π2n−2isinπ2ncosπ2n=−2isinπ2ncosπ2n+isinπ2n Thus, 1+ω+ω2+…+ωn−1=2−2isinπ2ncosπ2n+isinπ2n=icosπ2n−isinπ2nsinπ2n=icosπ2nsinπ2n−i2sinπ2nsinπ2n=1+icot(π2n)