What is $\arctan(k\cdot\tan(x))$? Is there any simplification for this?

Question

What is $\arctan(k\cdot\tan(x))$ if $k$ is a real number? Is there any simplification for this? I want solve it interms of $x$. I would be happy if it was $x\cdot\arctan(k)$ but it is not. So a similar solution would help.

Thanks in advance.

Answer 1

There may not be an elementary answer, but you could Taylor expand to however many terms you need. For example, if $|k\cdot \tan (x)| < 1$ then

$$\arctan(k\cdot \tan(x)) \approx k\cdot \tan (x) - \frac{(k\cdot \tan (x))^3}{3} + \cdots$$

converges rather quickly. If instead $k\cdot \tan (x) > 1$, then use $\arctan(x) = \frac{\pi}{2} - \arctan(\frac{1}{x})$ to get

$$\arctan(k\cdot \tan(x)) \approx \frac{\pi}{2} - \frac{1}{k\cdot \tan (x)} + \cdots$$

A similar formula exists for large negative values, as well. All three converge quickly, if it's any consolation.