Consider the transfer function

Now, if I simply multiply by the conjugate I get this:

Separating the real part and the imaginary part, and taking the angle between them:

Now I know this should've been: , by simply inspecting the same function slightly rewritten:

Now , the numerator is obviously 90 degrees, and the denominator , so the answer would be .

Now why does this change by a simple rewrite and a different method? Are both equally "correct"?