Crazy Inverse Mistake

First let $tan^{-1}(x) = y$ .

Then, from the identity $tan^{-1}(x) + cot^{-1}(x) = \frac{\pi}{2}$ we have $cot^{-1}(x) = \frac {\pi}{2} - y$ .

Substituting into the original equation,

$(y)*(\frac {\pi}{2} - y)=1$

$\Rightarrow$ $y^2 -$ $\frac{\pi}{2}$ $y+1 = 0$

The above equation has no real roots for $y$ . Hence there are no real values of $tan^{-1}(x)$ satisfying the above equation. As the domain of $tan^{-1}(x)$ is all real numbers, so there is no real value of $x$ for which $tan^{-1}(x)$ is imaginary. Hence our answer is $\boxed{0}$