If z is a complex number of unit modulus and argument θ , then arg 1 + z 1 + z - can be equal to given z ≠ - 1

Step-by-step solution

Let z = cos   θ + i   sin   θ Then, 1 + z 1 + z ¯ = 1 + cos   θ + i   sin   θ 1 + cos   θ - i   sin   θ = 2   cos 2 θ 2 + i · 2   sin θ 2 cos θ 2 2   cos 2 θ 2 - i · 2   sin θ 2 cos θ 2 = cos θ 2 + i   sin θ 2 cos θ 2 - i   sin θ 2 = cos θ 2 + i   sin θ 2 2 = cos   θ + i   sin   θ ∴