Consider the transfer function \(\frac{1}{1-j\frac{w}{w_c}}\)

Now, if I simply multiply by the conjugate I get this:
\(\frac{1+j\frac{w}{w_c}}{1+\frac{w^2}{w_c^2}}\)

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

Now I know this should've been: \( \phi = 90 - arctan \frac{w}{w_c}\), by simply inspecting the same function slightly rewritten:
\(\frac{1}{1-j\frac{w}{w_c}} = \frac{j\frac{w}{w_c}}{1+j\frac{w}{w_c}}\)
Now , the numerator is obviously 90 degrees, and the denominator \(-\frac{w}{w_c}\), so the answer would be \( \phi = 90 - arctan \frac{w}{w_c}\).

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

 

The angle is:

Φ=arctan(ωc/ω)

which equals

Φ=90-arctan(ω/ωc)