Can I get Mathematica to recognize common series expansions of trigonometric functions?

Question

I have some complicated functions from numerical procedures that go to simple trig functions in the limit of small time-steps. For example, I can calculate that one of these functions goes to:

x - 2x^3 / 3 + 2x^5 / 15

This is $\sin{x} \cos{x}$ minus terms of $O(x^7)$ and higher. Is there any way for Mathematica to recognize such a pattern automatically? I have other similar series expansions and I don't want to have to find all of them manually.

---Edit---

Thank you Szabolcs and corey979! You asked for more examples/background so here it is. I wanted to convert a 2x2 matrix of polynomials to trigonomic functions. I can obtain this matrix to arbitrary order in some small parameter h. To first order that looks like:

{{1 + h (t - (2 t^3)/3 + (2 t^5)/15), h (t^2 - t^4/3 + (2 t^6)/45)}, {h (t^2 - t^4/3 + (2 t^6)/45), 1 + h (-t + (2 t^3)/3 - (2 t^5)/15)}}

To second order that looks like:

{{1 + h^2 (t^2/2 - t^4/6) + h (t - (2 t^3)/3 + (2 t^5)/15), h (t^2 - t^4/3 + (2 t^6)/45)}, {h (t^2 - t^4/3 + (2 t^6)/45), 1 + h^2 (t^2/2 - t^4/6) + h (-t + (2 t^3)/3 - (2 t^5)/15)}}

etc

For a single matrix element, I was able to collect the coefficients of various orders of h and use With the data and FindFunction approach to turn those coefficents into trigonomic functions. It seems to work okay if I have the coefficients have enough terms, but I can get pretty wacky numbers if it doesn't have enough terms. I use this code:

findtrigfunc[inseries_] :=  Module[{outseries=inseries},
outseries = outseries //CoefficientList[#,h]&;
For[i=1, i <= Length[outseries],i++, 
{data = Table[Evaluate@{x, outseries[[i]] //. t->x}, {x, 0, 1, 0.01}];
outseries[[i]] = FindFormula[data, x, TargetFunctions -> {Times,Sin, Cos}]}];
Sum[outseries[[i]]*h^(i- 1) , {i,1,Length[outseries]}]//.x->t]

Here's what that produces for the first matrix element up to second order in h and order 7 in t:

1. + 1.0043 h Cos[t] Sin[t] + h^2 (-0.00618535 + 0.596221 Sin[Sin[t]]^2)

When I increase the order in t from 7 to 11, I get extra terms that get me a more reasonable answer:

1. + h^2 (0.49996 - 0.499954 Cos[t]^2) + 1.00001 h Cos[t] Sin[t]

Answer 1

If you have sufficiently many terms, FindGeneratingFunction may be able to do exactly this. But I wasn't having much luck with it, so I went in a roundabout way:

This is the expression. The more terms you have the better it works.

expr = Normal@Series[Sin[x] Cos[x], {x, 0, 8}]
(* x - (2 x^3)/3 + (2 x^5)/15 - (4 x^7)/315 *)

Extract coefficients:

coeff = CoefficientList[expr, x]
(* {0, 1, 0, -(2/3), 0, 2/15, 0, -(4/315)} *)

Try FindGeneratingFunction. It doesn't work here. But if we had computed the series to order 12, it would work, and it would return 1/2 Sin[2 x] (try it).

FindGeneratingFunction[coeff, x]
(* FindGeneratingFunction[{0, 1, 0, -(2/3), 0, 2/15, 
  0, -(4/315)}, x] *)

Construct sequence and get rid of the factorials, in the hope that it leads to simpler terms:

seq = {#1, #1! #2} & @@@ Transpose@{Range[0, Length[coeff] - 1], coeff}
(* {{0, 0}, {1, 1}, {2, 0}, {3, -4}, {4, 0}, {5, 16}, {6, 0}, {7, -64}} *)

Now try FindSequenceFunction. When this method fails, this is where it will fail.

fun = FindSequenceFunction[seq]
(* -2^(-2 + #1) ((-I)^(1 + #1) + I^(1 + #1)) & *)

Re-sum the result:

Sum[fun[k]/k! x^k, {k, 0, Infinity}]
(* -(1/4) I E^(-2 I x) (-1 + E^(4 I x)) *)

Simplify:

FullSimplify[%]
(* Cos[x] Sin[x] *)

Answer 2

A similar idea to Szabolcs, but I was waiting for a few more examples from the OP. Here goes.

f[x_] := x - 2 x^3/3 + 2 x^5/15

Create some data from it

data = Table[Evaluate@{x, Normal@f[x]}, {x, 0, 0.1, 0.01}];

and try to FindFormula:

FindFormula[data, x, TargetFunctions -> {Times, Sin, Cos}]

Cos[x] Sin[x]

---

A few made-up examples:

g[x_] := Normal@Series[Tan[x] + Cos[x], {x, 0, 5}]
g[x]

1 + x - x^2/2 + x^3/3 + x^4/24 + (2 x^5)/15

data2 = Table[Evaluate@{x, Normal@g[x]}, {x, 0, 0.1, 0.01}];

FindFormula[data2, x, TargetFunctions -> {Times, Plus, Sin, Cos, Tan}]

1. Cos[x] + 1. Tan[x]

Also works fine.

---

h[x_] := Normal@Series[Sin[x] Tan[x], {x, 0, 5}]
h[x]

x^2 + x^4/6

data3 = Table[Evaluate@{x, Normal@h[x]}, {x, 0, 0.1, 0.01}];

FindFormula[data3, x, TargetFunctions -> {Times, Sin, Cos, Tan, Sec}]

0.999993 Sin[x] Tan[x]

Almost perfect (doesn't work without Sec though).

---

k[x_] := Normal@Series[Cos[x]^2 + Sin[x], {x, 0, 4}]
k[x]

1 + x - x^2 - x^3/6 + x^4/3

data4 = Table[Evaluate@{x, Normal@k[x]}, {x, 0, 0.1, 0.01}];

FindFormula[data4, x, TargetFunctions -> {Times, Plus, Sin, Cos}]

Cos[x]^2 + Sin[x]

---

Unfortunately, this fails

p[x_] := Normal@Series[Sin[x]^2 + Cos[x], {x, 0, 6}]
p[x]

1 + x^2/2 - (7 x^4)/24 + (31 x^6)/720

data5 = Table[Evaluate@{x, Normal@p[x]}, {x, 0, 0.1, 0.01}];

FindFormula[data5, x, TargetFunctions -> {Times, Plus, Sin, Cos}]

1 + 0.499998 x^2 - 0.29104 x^4

probably because Sin[x]^2 is small compared to Cos[x].