The possible values of $x$, if $\tan^{-1} x>\cot^{-1}x$ What are the possible values of $x$, if $\tan^{-1}x >\cot^{-1}x$?   
We have $\tan^{-1}x >\cot^{-1}x\implies \tan^{-1}x -\cot^{-1}x>0 \implies \tan^{-1} x-\tan^{-1}1/x>0\implies \tan^{-1}\dfrac{x-1/x}{1-x.1/x}>0$.
What can I do now?

Given $\displaystyle \tan^{-1}(x)>\cot^{-1}(x)\;\forall\; x\in\mathbb{R}.$
So  $\displaystyle \tan^{-1}(x)>\frac{\pi}{2}-\tan^{-1}(x)$
So $\displaystyle \tan^{-1}(x)>\frac{\pi}{4}\Rightarrow x>1$
So we get $x\in(1\;,\infty).$ 
