I am trying to translate the following equation in polar coordinates: $$ \left|\frac{e^{2jw} - 0.5}{e^{jw}(e^{jw}-0.5)} \right|^2$$
The result should be the following: $$\frac{1.25 - cos(2\omega)}{1.25+cos(\omega)}$$
Any help on how to do it? I managed to obtain it using $e^{jw} =cos(\omega) -jsen(\omega)$. I am sure there is a faster and easier way.
First, for any $\alpha \in \mathbb{R}$:
$$ \begin{align} \left|2 e^{j \alpha} - 1\right|^2 = \left(2 e^{j \alpha} - 1\right) \overline{\left(2 e^{j \alpha} - 1\right)} = \left(2 e^{j \alpha} - 1\right) \left(2 e^{-j \alpha} - 1\right) = 5 - 4 \operatorname{Re}\left(e^{j\alpha}\right) = 5 - 4 \cos \alpha \end{align} $$
Then, assuming $w \in \mathbb{R}$:
$$\left|\frac{e^{2jw} - 0.5}{e^{jw}(e^{jw}-0.5)} \right|^2=\left|\frac{2e^{2jw} - 1}{e^{jw}(2 e^{jw}-1)} \right|^2=\frac{\left|2e^{2jw} - 1\right|^2}{\left|e^{jw}\right|^2\cdot\left|2 e^{jw}-1\right|^2} = \frac{5 - 4 \cos 2w}{1 \cdot \left(5 - 4 \cos w\right)}=\cdots$$