What 's the differece between $\cot(x)$ and $\arctan(x)$? 
I know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$

What is the difference between these two function?
Is $\cot(x)$ the reciprocal function of $\space \tan(x) \space$ and $\arctan(x)$ is the inverse function of $\tan(x)$?
And, so the assumption that $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$, is incorrect?

Yes. $\cot x$ is the reciprocal, $\arctan x$ is the (principal) inverse,  and $\arctan x=\frac1{\tan x}$ is incorrect.
