Result of Reduce contains duplicate terms

Question

On my system, the result of this:

f[x_] := 2 Sin[x] + Sin[x]^2;
Reduce[f'[x] == 0, x]

contains the following expression twice:

x == -(Pi/2) + 2*Pi*C[1]

This is the entire result:

Element[C[1], Integers] && 
     (x == -(Pi/2) + 2*Pi*C[1] || x == Pi/2 + 2*Pi*C[1] || 
      x == -(Pi/2) + 2*Pi*C[1] || x == (3*Pi)/2 + 2*Pi*C[1])

Is there a reason the expression appears twice? Is this a bug?

Answer 1

I'll try to sum up here the answers given so far in comments:

• if you work in a single period, you get the expected results:

  In:=  Reduce[f'[x] == 0 && 0 <= x <= 2 Pi, x]
  Out=  x == \[Pi]/2 || x == (3 \[Pi])/2
  

• if you work on the whole real domain, you can get the expression to be reduced by using Simplify:

  In:=  Simplify[Reduce[f'[x] == 0, x]]
  Out=  C[1] \[Element] Integers && (\[Pi] + 2 x == 4 \[Pi] C[1] || \[Pi] + 4 \[Pi] C[1] == 2 x || 2 x == \[Pi] (3 + 4 C[1]))
  

• the reason why you get one of the elements twice in the first place is that it's a double root.