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Re: Re: Expanding powers of cosine

• To: mathgroup at smc.vnet.net
• Subject: [mg84210] Re: [mg84191] Re: [mg84184] Expanding powers of cosine
• From: Carl Woll <carlw at wolfram.com>
• Date: Fri, 14 Dec 2007 03:55:53 -0500 (EST)
• References: <200712130102.UAA23849@smc.vnet.net> <200712130626.BAA06862@smc.vnet.net>

Andrzej Kozlowski wrote:

>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
>  
>
Another method is to rely on TrigExpand and arc trigs:

In[175]:= TrigExpand[Cos[2 x]^4 /. x -> ArcSin[Sin[x]]]

Out[175]= 16 sin^8(x)-32 sin^6(x)+24 sin^4(x)-8 sin^2(x)+1

Carl Woll
Wolfram Research

• References:
  • Expanding powers of cosine
    • From: michael.p.croucher@googlemail.com
  • Re: Expanding powers of cosine
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>