How to further simplify and combine solution sets of trigonometric equations?

[mmamathes (Bill Blair)]

When solving the following trigonometric equation, the solution set obtained is:

Solve[{5 (Sin[5 x + \[Phi]] - Sin[x]) == 0, 
  0 <= \[Phi] <= 2 \[Pi]}, x, Reals]

How to further simplify and combine solution sets of trigonometric equations to obtain the following solution form?

{{x -> ConditionalExpression[
    1/6 (\[Pi] - \[Phi] + 
       2 \[Pi] ConditionalExpression[1, \[Placeholder]]), 
    ConditionalExpression[1, \[Placeholder]] \[Element] Integers && 
     0 <= \[Phi] <= 2 \[Pi]]}, {x -> 
   ConditionalExpression[
    1/4 (-\[Phi] + 2 \[Pi] ConditionalExpression[1, \[Placeholder]]), 
    ConditionalExpression[1, \[Placeholder]] \[Element] Integers && 
     0 <= \[Phi] <= 2 \[Pi]]}}

[mmamathes (Bill Blair)]

Is there no solution to this problem?

[gianlucagorni]

A simpler situation where this problem arises is for

Solve[Sin[x] == 0, x]

where Mathematica lists the odd and the even multiples of Pi separately. I don’t know how to merge them.

Somehow, I would expect the following to give True, but it does not:

first = Or[Element[x/2, Integers],
   Element[(x - 1)/2, Integers]];
second = Element[x, Integers];
Reduce[Equivalent[first, second], x, Reals]

[mroge02 (Michael Rogers)]

I don’t know a good way either, but maybe this can give you ideas:

Replace[FactorList[5 (Sin[5 x + \[Phi]] - Sin[x]), Trig -> True], {f_, _} :> 
    Solve[TrigReduce[f^2] == 0 /. 1 + Cos[q_] :> 1 - Cos[q + Hold[Pi]], x, 
     Reals], 1] // ReleaseHold // Apply@Join
(*
{{x -> ConditionalExpression[-(\[Phi]/4) + (Pi*C[1])/2, 
     Element[C[1], Integers]]}, 
  {x -> ConditionalExpression[-(Pi/6) - \[Phi]/6 + (Pi*C[1])/3, 
     Element[C[1], Integers]]}}
*)

It’s difficult to combine sequences in the original result, since the instances of C[1] are really independent variables. Then there are oddities like this:

Solve[{Cos[x] == 1}, x, Reals]
(*
{{x -> ConditionalExpression[2*Pi*C[1], Element[C[1], Integers]]}}
*)

Solve[{Cos[x] == -1}, x, Reals]
(* 
{{x -> ConditionalExpression[-Pi + 2*Pi*C[1], 
     Element[C[1], Integers]]}, 
  {x -> ConditionalExpression[Pi + 2*Pi*C[1], 
     Element[C[1], Integers]]}}
*)

The Hold[Pi] in my code gets around the problem with the last example.