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Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?

Can someone point me to a proof, or explain if it's a simple answer?

What I'm looking for is the point where it becomes understood that trigonometric functions and pi can be expressed as series. A lot of the information I find when looking for that seems to point back to arctan.

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    Based on the answers, perhaps what I'm really looking for is the proof that the derivative of arctan is $\frac{1}{1+x^2}$.2011-03-29
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    The derivative of $arctan x$ follows from: $$\begin{align}f \circ f^{-1} (x) = x &\Longrightarrow f'(f^{-1}(x)) \cdot (f^{-1})'(x) = 1 \\ &\Longrightarrow (f^{-1})'(x) = 1 /f'(f^{-1}(x)) \end{align}$$ (assuming all terms appearing _exist_ of course. Now use $f(x) = \tan x$.2011-03-29
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    This video cleared it up for me. http://www.youtube.com/watch?v=tky25AUK7Io2011-03-29
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    That $\arctan'(x)=1/(1+x^2)$ may be a definition. That depends on what you chose as a starting point for the construction of so-called *"elementary functions"*. One such construction starts with the complex exponential, another starts from the definition of arctan and log as antiderivatives of respectively $1/(1+x^2)$ and $1/x$. There are still other ways to proceed.2015-04-13

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