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

 From: Bill Page Subject: Re: [Axiom-developer] unexpected behaviour of normalize(1-(cos(x))^2) Date: Thu, 13 Aug 2009 00:57:22 -0400

```Why not use removeCosSq ?

(3) -> removeCosSq(f)

2
(3)  sin(x)
Type: Expression(Integer)
(4) -> removeCosSq(f+3)

2
(4)  sin(x)  + 3
Type: Expression(Integer)
(5) -> removeCosSq(f+a)

2
(5)  sin(x)  + a
Type: Expression(Integer)

(6) -> removeCosSq(2*(f+a))

2
(6)  2sin(x)  + 2a
Type: Expression(Integer)
(7) -> removeCosSq(1/(f+a))

1
(7)  -----------
2
sin(x)  + a
Type: Expression(Integer)

removeCoshSq
removeSinSq
removeSinhSq

Regards,
Bill Page.

On Wed, Aug 12, 2009 at 10:47 PM, Michael Becker wrote:
> Am Mittwoch, 5. August 2009 15:59 schrieben Sie:
>> Michael,
>>
>> 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
>>
>>   sinrule(E)
>
>
>
>
>    Tim,
>
>
>    this does not always  work (see (6) and (7)) :
>
>
>
> (1) -> )set mess auto off
> (1) ->  sinrule:=rule((1-cos(x)^2) == sin(x)^2)
> (1) ->
>                2                   2
>   (1)  - cos(x)  + %C + 1 == sin(x)  + %C
>                        Type: RewriteRule(Integer,Integer,Expression Integer)
> (2) -> f:= 1 - cos(x)^2
> (2) ->
>                2
>   (2)  - cos(x)  + 1
>                                                     Type: Expression Integer
> (3) -> sinrule(f)
> (3) ->
>              2
>   (3)  sin(x)
>                                                     Type: Expression Integer
> (4) -> sinrule(f+3)
> (4) ->
>                2
>   (4)  - cos(x)  + 4
>                                                     Type: Expression Integer
> (5) -> sinrule(f+a)
> (5) ->
>              2
>   (5)  sin(x)  + a
>                                                     Type: Expression Integer
> (6) -> sinrule (2*(f+a))
> (6) ->
>                 2
>   (6)  - 2cos(x)  + 2a + 2
>                                                     Type: Expression Integer
> (7) -> sinrule (1/(f+a))
> (7) ->
>                 1
>   (7)  - ---------------
>                2
>          cos(x)  - a - 1
>                                                     Type: Expression Integer
>
>
>
>
>    - Michael
>
>
>
>
>
>>
>> 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:
>> >     Hi,
>> >
>> >
>> >    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
>> >                 4tan(-)
>> >                      2
>> >    (30)  ----------------------
>> >              x 4        x 2
>> >          tan(-)  + 2tan(-)  + 1
>> >              2          2
>> >                                                      Type: Expression
>> > Integer
>> >
>> >
>> >
>> >
>> >
>> >
>> >     -- Michael
>
>
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