Question

If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).

Answer 1

We have `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy

⇒ `[(1 + i)/(1 - i) xx (1 + i)/(1 + i)]^3 - [((1 - i)(1 - i))/((1 + i)(1 - i))]^3` = x + iy

⇒ `[(1 + i^2 + 2i)/(1 - i^2)]^3 - [(1 + i^2 - 2i)/(1 - i^2)]^3` = x + iy

⇒ `[(1 - 1 + 2i)/(1 + 1)]^3 - [(1 - 1 - 2i)/(1 + 1)]^3` = x + iy

⇒ `((2i)/2)^3 - ((-2i)/2)^3` = x + iy

⇒ (i)³ – (–i)³ = x + iy

⇒ i².i + i².i = x + iy

⇒ –i – i = x + iy

⇒ 0 – 2i = x + iy

Comparing the real and imaginary parts,

We get x = 0, y = –2

Hence, (x, y) = (0, –2).