[?] Calculate a trigonometric function?

Questions
1

Please help me with calculating the following trigonometric function

ArcTan[1/x] + ArcTan[x]]

Thanks.

2

Reduce[z == ArcTan[1/x] + ArcTan[x], {x}, Reals] gets you most of the way there.

This seems to do the trick:

PiecewiseExpand@Piecewise[
  Join @@ 
     With[{assum = And @@ Cases[#, Except[_Equal]]},
       ({z, assum} /. Solve[FullSimplify[#, assum]])] & /@ 
   List @@ LogicalExpand@
     Reduce[z == ArcTan[1/x] + ArcTan[x], {x}, Reals],
  Indeterminate]  (* <-- or Undefined *)