​ If (1+i/1- i)^93-(1- i/1+i)^3 = x + iy, find x and y. ​

Question

If \((\frac{1+i}{1-i})^{93}-(\frac{1-i}{1+i})^{3}\) = x + iy, find x and y.

Answer 1

In denominator, we use the identity

(a – b)(a + b) = a² – b²

= (i)⁹³ – (-i)³ = (i)⁹²+1 – [-(i)³] = [(i)⁹²(i)] – [-(i² × i)] = [(i⁴)²³(i)] – [-(-i)] = [(1)²³(i)] – i = i - i

x + iy = 0

∴ x = 0 and y = 0