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Re: Expanding powers of cosine

  • To: mathgroup at smc.vnet.net
  • Subject: [mg84191] Re: [mg84184] Expanding powers of cosine
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 13 Dec 2007 01:26:39 -0500 (EST)
  • References: <200712130102.UAA23849@smc.vnet.net>

On 13 Dec 2007, at 10:02, michael.p.croucher at googlemail.com wrote:

> Hi
>
> I would like to express even powers of Cos[x] in terms of powers of
> Sin[x] using the identity Sin[x]^2+Cos[x]^2 = 1.  For example
>
> Cos[x]^4 = 1 - 2 Sin[x]^2 + Sin[x]^4
>
> I could not get any of Mathematica's built in functions to do this for
> me so I created my own rule:
>
> expandCosn[z_] := Module[{s, res},
>  s = Cos[x]^n_ :> (1 - Sin[x]^2) Cos[x]^(n - 2) ;
>  res = z //. s;
>  Expand[res]
>  ]
>
> which works fine:
>
> In[14]:= expandCosn[Cos[x]^4]
>
> Out[14]= 1 - 2 Sin[x]^2 + Sin[x]^4
>
> My question is - have I missed something?  Is there an easier way to
> do this?
>
> Cheers,
> Mike
>

Here is one way. This is how to expand Cos[x]^24:

First[GroebnerBasis[{Cos[x]^24, 1 - Cos[x]^2 - Sin[x]^2}, {Sin[x]},
      {Cos[x]}]]

  Sin[x]^24 - 12*Sin[x]^22 + 66*Sin[x]^20 -
    220*Sin[x]^18 + 495*Sin[x]^16 - 792*Sin[x]^14 +
    924*Sin[x]^12 - 792*Sin[x]^10 + 495*Sin[x]^8 -
    220*Sin[x]^6 + 66*Sin[x]^4 - 12*Sin[x]^2 + 1

Andrzej Kozlowski


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