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Re: Re: Wrong Simplify[] Answer for

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104459] Re: [mg104410] Re: [mg104400] Wrong Simplify[] Answer for
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Sun, 1 Nov 2009 03:59:21 -0500 (EST)
  • References: <200910300719.CAA27787@smc.vnet.net>
  • Reply-to: drmajorbob at yahoo.com

In other words,

one = Cos[x]^4 - Sin[x]^4 // Factor

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

two = MapAt[Simplify, one, {-1}]

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

three = two // Expand

Cos[x]^2 - Sin[x]^2

four = three // Simplify

Cos[2 x]

The last step is the double-angle formula, which is easily proven from  
Euler's formula:

Exp[x I] Exp[y I] // ComplexExpand
% /. y -> x
Exp[2 x I] // ComplexExpand

Cos[x + y] + I Sin[x + y]

Cos[2 x] + I Sin[2 x]

Cos[2 x] + I Sin[2 x]

(equating real and imaginary parts in the first and last result)

Bobby

On Sat, 31 Oct 2009 01:49:11 -0500, Pratip Chakraborty  
<pratip.chakraborty at gmail.com> wrote:

> 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
>
> On Fri, Oct 30, 2009 at 8:19 AM, Lawrence Teo <lawrenceteo 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.
>>
>>
>


-- 
DrMajorBob at yahoo.com


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