The value of ∑ k = 1 4 s i n 2 π k 5 - i c o s 2 π k 5 4 is (where i is iota)

Step-by-step solution

∑ k = 1 4 s i n 2 π k 5 - i c o s 2 π k 5 ∑ k = 1 4 - i 2 s i n 2 π k 5 - i c o s 2 π k 5 = - i ∑ k = 1 4 e i 2 π k 5 = - i - 1 + e i 0 + e i 2 π 5 + e i 4 π 5 + e i 6 π 5 + e i 8 π 5 (Sum of roots of the fifth root of unity is zero) = i ⇒ ∑ k = 1 4 ( s i n 2 π k 5 - i c o s 2 π k 5 4 = i 4 = 1