Suppose z is a complex number such that z≠−1, |z|=1, and  arg⁡(z)=θLet ω =z(1−z¯)z¯(1+z),than Re⁡(ω) ) is equal to

ω=z−zz¯z¯+z¯z=z−11/z+1=(z−1)zz+1 [∵|z|=1]=z2−1z+1−z−1z+1=z−1+1−z1+zRe⁡(ω)=Re⁡(z)−1+Re⁡1−z1+zBut Re⁡1−z1+z=121−z1+z+1−z1+z=121−z1+z+z−1z+1=0∴ Re⁡(ω)=cos⁡θ−1=−2sin2⁡(θ/2)