If α  is the fifth root of unity then

we have α=cos2π5+isin2π5 and 1+α+α2+α3+α4=0                ∴|1+α+α2+α3|=|−α4|=|α4|=1 Also, |1+α+α2|=|−α3(1+α)|=|1+α|...................(i)                                =|1+cos2π5+isin2π5|                                =|2cosπ5(cosπ5+isinπ5)|=2cosπ5 Again from (i), |1+α|=|1+α+α2|=2cosπ5