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?

HINT:
$$\cot^{-1}x=\frac\pi2-\tan^{-1}x\implies \tan^{-1}x-\cot^{-1}x=2\tan^{-1}x-\frac\pi2>0$$
