I have searched for this for a while, but I cannot find an explicit answer to the following.

Are there any interesting properties for the following:

* $\tan^{-1}(x+y)$
* $\tan^{-1}(x\cdot y)$
* $\tan^{-1}(\frac{x}{y})$
* $\cos(\tan^{-1}(x))$
* $\sin(\tan^{-1}(x))$
* $\frac{1}{\tan^{-1}(x)}$

The context of why I am asking for this is because I have 2 known functions $f\_0(t)$ and $f\_1(t)$ and I am trying to simplify the expressions:

$$\theta=\tan^{-1}\Big(\frac{f\_1(t)}{f\_0{t}}\Big)$$

$$r=\frac{f\_0(t)}{\cos\Big(\tan^{-1}\Big(\frac{f\_1(t)}{f\_0{(t)}}\Big)\Big)}$$

Which are a representation of a straight line in polar coordinates (for the correct $f\_0$ and $f\_1$)