I've been trying to use the $|z|^2 = z\overline{z}$ to transform the identity and then just compare the sides but I end up having something inequal. What's a good and efficient solution to this?
Identity to prove:
$$|1-\overline{a}b|+|a-b|^2 = (1+|ab|)^2 - (|a|^2 + |b|^2)$$
After some answers/comments I've realized that the identity is wrongly written in the book. With a tiny edit in the text, it can be proven easily. Thanks for the help! (Answers given by: Lord Shark The Unknown and amsmath.