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Re: Wrong Simplify[] Answer for Simplify[Cos[x]^4-Sin[x]^4]?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104426] Re: Wrong Simplify[] Answer for Simplify[Cos[x]^4-Sin[x]^4]?
  • From: Helen Read <hpr at together.net>
  • Date: Sat, 31 Oct 2009 01:52:29 -0500 (EST)
  • References: <hce437$r4t$1@smc.vnet.net>

Lawrence Teo wrote:
> We know that Simplify[Cos[x]^2-Sin[x]^2] -> Cos[2 x]
> But why Simplify[Cos[x]^4-Sin[x]^4] -> Cos[2 x] too?
> 
> Doing subtraction between the two expressions will give small delta.
> This is enough to prove that the two expression shouldn't be the same.
> 
> Can anyone give me any insight? Thanks.

The two expressions are in fact equal.

Cos[x]^4 - Sin[x]^4 factors into

(Cos[x]^2 + Sin[x]^2)(Cos[x]^2 - Sin[x]^2)

and Cos[x]^2 + Sin[x]^2 == 1 (Pythagorean Identity)

hence Cos[x]^4 - Sin[x]^4 == Cos[x]^2 - Sin[x]^2

QED

--
Helen Read
University of Vermont



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