absolute value of sum of complex numbers squared

Question

is this correct

$ \left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-2|a||b|+|b|^2$

Thank you

Answer 1

$$\left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=(|a| \exp(-i c)-|b| \exp(-i d) )\times( |a| \exp(i c)-|b| \exp(i d))$$ so $$\left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-(\exp(i (c-d))+\exp(i (d-c))|a||b| +|b|^2$$ so $$\left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-2\cos(c-d)|a||b| +|b|^2$$