What are the algebraic properties of the arctan function?

Question

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 $f_1$)

Answer 1

Your third and fourth expressions are standard problems in high school trigonometry. One intuitive way to solve these is to use this diagram:

You can see here that

$$\theta = \tan^{-1}(x)$$

So using the clear expressions for $\cos\theta$ and $\sin\theta$ from the diagram we get

$$\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{x^2+1}}$$

$$\sin(\tan^{-1}(x)) = \frac{x}{\sqrt{x^2+1}}$$

With a little thought, you can see that this does not just apply for positive $x$ but for all real values $x$. That is confirmed with simple graphing:

Your other expressions are not so simple.

Answer 2

None of these functions have particularly nice forms. You might have success using the definition of arctan in terms of the logarithm (which can be easily constructed based on the formula for tan using $e^x$) and simplifying, if your functions involve exponents.