2011/102) Prove z/ (1 + z^2) is real if and only if z is real or |z| =1.

if z is zero it is true so let us take for the case z not zero

so z/(1+z^2) = 1/(z + 1/z) is real

let z = r cos t + ir sin t

so 1/z = 1/r cos t - 1/r i sin t

z + 1/z imaginary part is zero

r sin t = - 1/r sin t
sin t = 0 or r = -1/r or |r| = 1 and hence |z| = 1


sin t = 0 means real

hence z is real or |z| = 1