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