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

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

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

• How about $\sqrt[\cot(x)]{e}?$ – Gummy bears Sep 3 '15 at 16:36

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.
• No problem; it's a happy coincidence that you posted this question the same day I used just such a method to answer another question. Additionally, it's just occurred to me that the infinite product might be useful in this way: we can expand each term in the product as a Maclaurin series in powers of $x$ (rather than $\tan{x}$), and use the infinite product to calculate a Taylor series for $\exp{\tan{x}}$ in powers of $x$. – Will R Sep 3 '15 at 17:35
What about $$\sqrt[\cos(x)]{e^{\sin(x)}}?$$