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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104409] Re: Wrong Simplify[] Answer for Simplify[Cos[x]^4-Sin[x]^4]?
  • From: pratip <pratip.chakraborty at gmail.com>
  • Date: Sat, 31 Oct 2009 01:49:00 -0500 (EST)
  • References: <hce437$r4t$1@smc.vnet.net>

On Oct 30, 8:20 am, Lawrence Teo <lawrence... at yahoo.com> 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.

Hi,
Please remember the basic identity
        Cos[x]^2+Sin[x]^2=1   (* We multiply both sides of the
equation with (Cos[x]^2-Sin[x]^2) *)
        =>(Cos[x]^2+Sin[x]^2)*(Cos[x]^2-Sin[x]^2)=1*(Cos[x]^2-Sin[x]
^2)   (* remember (a+b)(a-b)=a^2-b^2 *)
        =>(Cos[x]^4-Sin[x]^4)=Cos[2x]
Also for this type of doubt one can take help of the Plot function in
Mathematica.

Plot[Evaluate[{Cos[x]^4 - Sin[x]^4, Cos[2 x],
   Cos[x]^2 - Sin[x]^2}], {x, -2 Pi, 2 Pi},
 PlotStyle -> {{Red}, {Blue, Dashed}, {Cyan}}]

You will see all the three functions that we are plotting will
coincide.
Hope this helps you.

Regards,
Pratip


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