What other form can $\exp(\tan x)$ be written as.

Question

Is there a another form in which $\exp(\tan x)$ can be written as?

For example can it be written as: $e^{\tan x}$.

Answer 1

You could expand $\exp{(\tan{(x)})}$ as a Taylor-like polynomial in $\tan{(x)}$; this is handy in certain integration problems, like this one. This is valid for all $x$ for which $\tan{(x)}$ is defined.

Alternatively, you could begin by expanding $\tan{(x)}$ as a Taylor series for $|x|<\pi/2$, as given here on Wikipedia. Then you could express $e^{\tan{(x)}}$ as an infinite product of exponentials, using $e^{a}e^{b}=e^{a+b}$. I'm not sure what uses, if any, this approach might have.

Answer 2

What about $$\sqrt[\cos(x)]{e^{\sin(x)}}?$$

Answer 3

There are plenty of ways to write any function in terms of other functions. For example you can use Taylor series expansion or fourier transforms to represent any functions given that you have continues set of derivatives.