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Re: [Axiom-developer] unexpected behaviour of normalize(1-(cos(x))^2)

From: Tim Daly
Subject: Re: [Axiom-developer] unexpected behaviour of normalize(1-(cos(x))^2)
Date: Wed, 05 Aug 2009 09:59:59 -0400
User-agent: Thunderbird (Windows/20090302)


Trig identity substitutions are somewhat problematic in Axiom.
See the src/input/schaum* files for examples.

If the subexpression (1-cos(x)^2) occurs in your expression E you can write:

  sinrule:=rule((1-cos(x)^2) == sin(x)^2)

and then use this rule for your expression E thus


Axiom will not derive several of the trig identities from scratch.

In your expression we have something of the form
   (4a^2) / (a^2 + 1)^2    where a = tan(x/2)
so Axiom needs to show that
  (a^2+1)^2 != 0
  (a^2+1) != 0
  a^2 != -1
  a != i
or, by back-substitution
 tan(x/2) != i
which it does not conclude automatically, even though this
is clearly true in the domain Expression(Integer).

Michael Becker wrote:

   Is this (30)  the expected bevaviour of 'normalize' ??

(29) -> normalize ((sin(x))^2+(cos(x))^2)
(29) ->
   (29)  1
                                                     Type: Expression Integer

(30) -> normalize (1-(cos(x))^2)
(30) ->
                     x 2
   (30)  ----------------------
             x 4        x 2
         tan(-)  + 2tan(-)  + 1
             2          2
                                                     Type: Expression Integer

-- Michael

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