Find the angle of complex number Let
$$
z = \frac{a-jw}{a+jw}.
$$
Then the angle of $z$ is
$$
-\tan^{-1}z\left(\frac{w}{a}\right) -\tan^{-1}\left(\frac{w}{a}\right) = -2\tan^{-1}\left(\frac{w}{a}\right).
$$
How is that so?

Assuming $a,w\in\mathbb{R}$:
$$\arg\left(\frac{a-wi}{a+wi}\right)=$$
$$\arg\left(\frac{a-wi}{a+wi}\cdot\frac{a-wi}{a-wi}\right)=$$
$$\arg\left(\frac{(a-wi)^2}{a^2+w^2}\right)=$$
$$\arg\left(\frac{a^2-w^2-2awi}{a^2+w^2}\right)=$$
$$\arg\left(a^2-w^2-2awi\right)-\arg\left(a^2+w^2\right)=$$
$$\arg\left(a^2-w^2-2awi\right)-0=$$
$$\arg\left(a^2-w^2-2awi\right)=$$
$$\arg\left(a^2-w^2-2awi\right)=$$
$$\arg\left((a-wi)^2\right)=$$
$$\arg\left((a-wi)(a-wi)\right)=$$
$$\arg\left(a-wi\right)+\arg\left(a-wi\right)=$$
$$2\arg\left(a-wi\right)=$$
$$-2\tan^{-1}\left(\frac{w}{a}\right)$$
