From: kogeddes@daisy.uwaterloo.ca (Keith O. Geddes)
Subject: Re: Maple simplify Problem 
Organization: University of Waterloo
Date: Tue, 15 Dec 1992 14:02:37 GMT

In article <gwS=8F@engin.umich.edu> amadi@engin.umich.edu (Amadi Nwankpa) writes:
>     Maple seems to have a problem with this simplification:
>
>> ad;
>
>      {z(x) = 1/2 exp(x) + 1/2 exp(- x), y(x) = 1/2 exp(x) - 1/2 exp(- x)}
>
>> simp2;
>
>           {cosh(x) + sinh(x) = exp(x), cosh(x) - sinh(x) = exp(- x)}
>
>>  simplify(ad,simp2,[exp(x),exp(-x)]);
>                              2                              2
>                   1 + sinh(x)  + 2 sinh(x) cosh(x) + cosh(x)
>           {z(x) = -------------------------------------------,
>                              2 sinh(x) + 2 cosh(x)
>
>                                   2                              2
>                      - 1 + sinh(x)  + 2 sinh(x) cosh(x) + cosh(x)
>               y(x) = ---------------------------------------------}
>                                  2 sinh(x) + 2 cosh(x)
>
>> simplify(");
>
>                       {y(x) = - sinh(x), z(x) = cosh(x)}
>
>
>   If you put x = 1  and do a evalf you will see that the last output doesn't
>equal the value of ad for x =1 .  What did I do wrong?
>
>                                       Amadi Nwankpa
>                                       CAEN
>                                       Umich of Michigan


Yes, the result of that final simplify command should be

                      {y(x) = sinh(x), z(x) = cosh(x)}

You have discovered a bug in one line of the library file `simplify/trig`
where it thinks that  sinh(x)^2 = cosh(x)^2 + 1  rather than the correct
transformation  sinh(x)^2 = cosh(x)^2 - 1 .

I will send you the corrected file for `simplify/trig` and we will get
this bug fix placed into the share library for ftp access.

Here is the result of running your example after reading in the corrected
file for `simplify/trig`.

==============================================================================

    |\^/|     Maple V Release 2
._|\|   |/|_. Copyright (c) 1981-1992 by the University of Waterloo.
 \  MAPLE  /  All rights reserved. MAPLE is a registered trademark of
 <____ ____>  Waterloo Maple Software.
      |       Type ? for help.

> read `simplify.trig.new`;

> ad := {z(x) = 1/2*exp(x) + 1/2*exp(-x), y(x) = 1/2*exp(x) - 1/2*exp(-x)};
   ad := {z(x) = 1/2 exp(x) + 1/2 exp(- x), y(x) = 1/2 exp(x) - 1/2 exp(- x)}

> simp2 := {cosh(x) + sinh(x) = exp(x), cosh(x) - sinh(x) = exp(-x)};
      simp2 := {cosh(x) + sinh(x) = exp(x), cosh(x) - sinh(x) = exp(- x)}

> simplify(ad,simp2,[exp(x),exp(-x)]);
                              2                              2
                   1 + cosh(x)  + 2 cosh(x) sinh(x) + sinh(x)
           {z(x) = -------------------------------------------,
                              2 cosh(x) + 2 sinh(x)

                                   2                              2
                      - 1 + cosh(x)  + 2 cosh(x) sinh(x) + sinh(x)
               y(x) = ---------------------------------------------}
                                  2 cosh(x) + 2 sinh(x)

> simplify(");
                        {z(x) = cosh(x), y(x) = sinh(x)}

> quit