In the comments section of [another post](https://math.stackexchange.com/questions/1707275/how-i-can-decompose-ln3fx2gy/), MATHEMATIKER stated that

$$\log(a+b)=\log(a)-2\log\left(\cos\left(\arctan\left(\sqrt{\frac{b}{a}}\right)\right)\right)$$

if $b>a>0$.

I wish to know how this was derived, and if it can be extended to $b,a\in\mathbb{C}$.