       Re: Re: Wrong Simplify[] Answer for

• To: mathgroup at smc.vnet.net
• Subject: [mg104570] Re: [mg104513] Re: Wrong Simplify[] Answer for
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Wed, 4 Nov 2009 01:34:33 -0500 (EST)

```a = Cos[x]^2 - Sin[x]^2;
b = Cos[x]^4 - Sin[x]^4;

Table[a - b, {x, -10, 10}] // Simplify

{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}

Note what happens when you use precision greater than machine precision

Off[N::meprec]

Table[a - b, {x, -10, 10}] // N[#, 20] &

{0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-250,0.*10^-250,0.*10^-250,0,0.*10^-250,0.*10^-250,0.*10^-250,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249,0.*10^-249}

Bob Hanlon

---- Lawrence Teo <lawrenceteo at yahoo.com> wrote:

=============
Hi all,

Thanks for the insight. So Simplify[] in Mathematica is right.
But why I observe small delta if I subtract the two expressions with // N?
Is it because of machine precision related limitation?

a = Cos[x]^2 - Sin[x]^2
b = Cos[x]^4 - Sin[x]^4
Table[a - b, {x, -10, 10}] // N

Return small delta...
\!\({6.938893903907228`*^-17, 6.245004513516506`*^-17, 0.`,
0.`, 7.025630077706069`*^-17, 0.`, 0.`, 3.854338723185968`*^-17,
0.`,
1.1102230246251565`*^-16, 0.`, 1.1102230246251565`*^-16, 0.`, \
3.854338723185968`*^-17, 0.`, 0.`, 7.025630077706069`*^-17, 0.`, 0.`,
\
6.245004513516506`*^-17, 6.938893903907228`*^-17}\)

On Oct 31, 2:50 pm, Peter Breitfeld <ph... at t-online.de> wrote:
> 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.
>
> it's simply true, because:
>
> cos^4 x - sin^4 x = (cos^2 x - sin^2 x)(cos^2 x + sin^2 x)
> = (cos^2 x - sin^2 x) * 1 = cos 2x
>
> --
> _________________________________________________________________
> Peter Breitfeld, Bad Saulgau, Germany --http://www.pBreitfeld.de

```

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