# Impose an specific Trigonometric expansion with TrigExpand

Is there a way to impose in mathematica an specific trigonometric expansion with TrigExpand?

I have various double-angle expressions for cosine and sine in my problem and i need to rewrite them in terms of cosines and sines functions. With this code you can get the same expressions than me.

ClearAll["Global*"]
SetOptions[$FrontEndSession, PrintingStyleEnvironment -> "Condensed"]$PrePrint = # /. {Csc[z_] :> 1/Defer@Sin[z],
Sec[z_] :> 1/Defer@Cos[z],
Cot[z_] :> Defer@Cos[z]/Defer@Sin[z]} &;
Derivative /:
Module[{bb, dd, sp},
MakeBoxes[dd, _] ^=
If[Length[{\[Alpha]}] == 1, "\[DifferentialD]", "\[PartialD]"];
MakeBoxes[sp, _] ^= "\[ThinSpace]";
bb /: MakeBoxes[bb[x__], _] := RowBox[Map[ToBoxes[#] &, {x}]];
TemplateBox[{ToBoxes[bb[dd^Plus[\[Alpha]], f1]],
ToBoxes[Apply[bb,
Riffle[Map[bb[dd, #] &,
Select[({vars}^{\[Alpha]}), (# =!= 1 &)]], sp]]],
ToBoxes[Derivative[\[Alpha]][f1][vars]]}, "ShortFraction",
DisplayFunction :> (FractionBox[#1, #2] &),
InterpretationFunction :> (#3 &), Tooltip -> Automatic]]

Subscript[w, 1] =
f[Subscript[r, 1], Subscript[r, 2], \[Alpha], \[Beta], \[Gamma],
Subscript[\[CapitalTheta], 12]];
Set[J, -(D[Subscript[w, 1], {\[Beta], 2}] +
1/Sin[\[Beta]]^2 (D[Subscript[w, 1], {\[Alpha], 2}] +
D[Subscript[w, 1], {\[Gamma], 2}]) +
Cot[\[Beta]] D[Subscript[w, 1], \[Beta]] -
2 Cot[\[Beta]]/
Sin[\[Beta]] D[Subscript[w, 1], \[Alpha], \[Gamma]])] //
Set[Subscript[a, 1],
2 D[Subscript[w, 1], {\[Beta], 2}] +
D[Subscript[w, 1], {\[Gamma], 2}] + J] // TraditionalForm ;
Set[Subscript[a, 2],
2 Cot[\[Beta]]/
Sin[\[Beta]] D[Subscript[w,
1], {\[Alpha], 1}] - (1 + 2 Cot[\[Beta]]^2) D[Subscript[w,
1], {\[Gamma], 1}] -
2/Sin[\[Beta]] D[Subscript[w, 1], \[Beta], \[Alpha]] +
2 Cot[\[Beta]] D[Subscript[w, 1], \[Beta], \[Gamma]]] //
Set[Subscript[f, 1],
1/(2 Sin[Subscript[\[CapitalTheta], 12]]^2) (-J +
Cos[Subscript[\[CapitalTheta],
12]] (Sin[2 \[Gamma]] Subscript[a, 2] -
Cos[2 \[Gamma]] Subscript[a, 1]) +
Sin[Subscript[\[CapitalTheta],
12]] (Sin[2 \[Gamma]] Subscript[a, 1] +
Cos[2 \[Gamma]] Subscript[a, 2])) -
D[Subscript[w, 1], Subscript[\[CapitalTheta], 12], \[Gamma]] -
1/2 Cot[Subscript[\[CapitalTheta], 12]] D[Subscript[w,
1], {\[Gamma], 1}] + (1/4 - 1/(
2 Sin[Subscript[\[CapitalTheta], 12]]^2)) D[Subscript[w,
1], {\[Gamma], 2}] ] // ExpandAll // TraditionalForm ;
Set[lapla1, -(1/(
2 Subscript[\[Mu], 1])) (1/Subscript[r, 1]
D[Subscript[w, 1] Subscript[r, 1], {Subscript[r, 1], 2}] -
1/Subscript[r,
1]^2 (D[Subscript[w,
1], {Subscript[\[CapitalTheta], 12], 2}] +
Cot[Subscript[\[CapitalTheta], 12]] D[Subscript[w,
1], {Subscript[\[CapitalTheta], 12], 1}]) - Subscript[f,
1]/Subscript[r, 1]^2)] // ExpandAll // TraditionalForm


In this specific case, when I use TrigExpand for lapla1, mathematica does not use a unique relation for doing the expansion of each of the terms and is quite difficult to follow it by hand. So, I would like to impose these two specific trigonometric expansions in order to check the results.

$$\sin(2x)= 2\cos x \sin x$$ $$\cos(2x)= \cos^2 x-\sin ^2x$$

Yo can check that those expressions above can be written in several ways:

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Each question should be standalone, so please include the expression in your question that you would like to simplify. – RunnyKine Mar 31 '14 at 16:02
@RunnyKine I edited my question. Sorry for any caused trouble. – shadraws Mar 31 '14 at 16:21
I was using wrong the TrigExpand function, so what I did to solve the problem was lapla1 /. {Sin[2 \[Gamma]] -> 2 Sin[\[Gamma]] Cos[\[Gamma]] , Cos[2 \[Gamma]] -> Cos[\[Gamma]]^2 - Sin[\[Gamma]]^2} // ExpandAll // TraditionalForm. Sorry for any inconvenience – shadraws Mar 31 '14 at 17:53

One way would be like the following. Let us define the function rule as follows:

Clear[rule];
rule[expr_] := ReplaceAll[ expr, {Sin[2 γ_] -> 2*Sin[γ]*Cos[γ],
Cos[2 γ_] -> Cos[γ]^2 - Sin[γ]^2}];


and map this function on your expression. For the sake of shortness I take here only a small part of your otherwise a too long expression. The effect is, however, the same, I checked. So, let this:

 expr=(Cos[Subscript[Θ, 12]] Sin[2 γ] f[Subscript[r, 1], Subscript[r, 2], α, β, γ,
Subscript[Θ, 12]])/(2 Sin[β] Sin[ Subscript[Θ, 12]]^2
\!$$\*SubsuperscriptBox[\(r$$, $$1$$, $$2$$]\) Subscript[μ, 1])


be your expression in the StandardForm. Then this

Map[rule, expr]//TraditionalForm


returning this:

(*  (sin(γ) cos(γ) cos(Subscript[Θ, 12]) f(Subscript[r, 1], Subscript[r, ],α,β,γ,
Subscript[Θ, 12]) )/(Subscript[μ, 1] Subsuperscript[r, 1, 2] sin(β)
sin(Subscript[Θ, 12])^2)  *)
`

Though it looks awfully here, this:

is what you see on the screen. And suchlike looks each term.

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