The point that I was making was not whether the method being used works - as it clearly does - but whether a student's response might be judged as showing all necessary working, especially in an assessment task where individual marking can be considerably more variable than HSC marking.
The method that you are using is, in effect, taking a problem on the Argand Diagram, converting it to an equivalent problem on the Cartesian Plane, solving it, and then translating that solution back to the Argand Diagram. Supporting those steps sufficiently calls for judgement from the student and immediate comprehension of a potentially unanticipated / unfamiliar approach from the marker. Whether you accept this as true or not, it is a fact that there is potential for an inadequately supported answer or a misinterpretation from the marker. I recognise that your use of vectors is actually staying with the Argand Diagram, but that will not be the case for many using this style of approach. Your method also converts the vector (1, 2.5
i) to (2.5, -
i) without explanation, which could be argued to be skipping necessary working.
I also suggest that your assertion that "the most superior students will not only know how to do each question, but they will also have identified the fastest and easiest way, this is because simple methods save time" is an oversimplification. There is not one unique "simplest" solution for every problem. A question might be solvable by an algebraic or geometric method, and Student A may find algebraic approaches simpler whilst Student B finds geometric reasoning simpler. Telling a class with Students A and B that one method is faster / easier / simpler disregards individual differences and competencies, which is potentially a disservice to both students and others in the class. Applying a longer method correctly and confidently may actually be faster for some students. For example, recognising and apply reasoning like
 \\ &= \sqrt{2}\left(\sin{x}\cos{\frac{\pi}{4}} + \cos{x}\sin{\frac{\pi}{4}}\right) \\ &= \sqrt{2}\sin{\left(x + \frac{\pi}{4}\right)} \end{align*})
is providing a solution more efficiently than if the full auxilliary angle method is used, but I would encourage a student to provide the full auxilliary angle method working and get to the correct answer if their attempts to use the short-cut does not reliably lead to the correct answer.
