Consider the transfer function $\displaystyle \frac{1}{1-j\frac{w}{w_c}}$

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

$\displaystyle \frac{1+j\frac{w}{w_c}}{1+\frac{w^2}{w_c^2}}$

Separating the real part and the imaginary part, and taking the angle between them: $\displaystyle \phi = arctan(b/a) = arctan \frac{\frac{w}{w_c}}{1+\frac{w^2}{w_c^2}} / \frac{1}{1+\frac{w^2}{w_c^2}} =arctan \frac{w}{w_c} $

Now I know this should've been: $\displaystyle \phi = 90 - arctan \frac{w}{w_c}$, by simply inspecting the same function slightly rewritten:

$\displaystyle \frac{1}{1-j\frac{w}{w_c}} = \frac{j\frac{w}{w_c}}{1+j\frac{w}{w_c}}$

Now , the numerator is obviously 90 degrees, and the denominator $\displaystyle -\frac{w}{w_c}$, so the answer would be $\displaystyle \phi = 90 - arctan \frac{w}{w_c}$.

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