Write -2 + i as x + i y ; r = (x^2 + y^2)^ (1/2)

Z = x + i y = r e^ i theta = r ( cos theta + i sin theta) from De Moivre;s

Theorem

where sin theta = y / (x^2 + y^2)^ (1/2) and

cos theta = x / (x^2 + y^2)^ (1/2)

Just substitute for Z + 1/Z and remember sin(-theta) = - sin theta.