I have solved the problem $|z-\frac{4}{z}|=2$ using the triangle inequality, as seen elsewhere on this site. However, I have also seen an alternative solution, where the user has simply squared the equation, giving $(z-\frac{4}{z})^2=4$.
Solving this gives the same answer of $1+\sqrt5$, so it clearly works, but I'm just not sure of the maths behind being able to use this approach. Is it just a coincidence, or is there something behind the method? I would definitely appreciate someone being able to explain the reasoning behind it.