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

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

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

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  • $\begingroup$ 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 $x$. $\endgroup$ – Will R Sep 3 '15 at 17:35
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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.

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

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