`to_poly_solve` inconsistency of solutions for equivalent forms of goniometric equations

You can use trigreduce to convert sin(x)/cos(x) to tan(x).  But I agree
that it would be nice if solve(sin(x)/cos(x)=1,x) gave a reasonable answer.
 Besides to_poly_solve, another way to handle trigonometric equations more
systematically is to put them in exponential form, e.g.
rectform(solve(exponentialize(...), ...)).  But the result can be ugly in
some cases.

As for to_poly_solve, it's unreasonable to expect in general that
equivalent solution sets will be syntactically identical. That is a very
hard problem (actually unsolvable in general), though over time, more and
more cases will be handled nicely.

            -s

On Wed, Apr 18, 2012 at 17:46, Duc Trung Ha <mathemage at gmail.com> wrote:

> Hola,

I am sorry if this would come as a completely stupid question but I was really

amazed at this Maxima's behavior:

When I searched for solution of goniometric equation `tan(x) = 1`::

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(%i22) to_poly_solve(tan(x) = 1, x);

                                                   %pi

                                    - 2 %pi %z30 - ---

                                                    2

(%o22)                %union([x = - ------------------])

                                            2

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

answer was given to me. On the other hand, when I entered what I consider an

equivalent form of the very same equation:

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(%i23) to_poly_solve((sin(x)/cos(x)) - 1 = 0, x);

                                                                   %pi

                                                    - 4 %pi %z40 - ---

                      %pi (8 %z40 + 1)                              2

(%o23) %union(%if(cos(----------------) # 0, [x = - ------------------],

                             4                              2

                                                                        3 %pi

                                           %union()), [x = 2 %pi %z38 - -----])

                                                                          4

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

output came out. After a brief inspection one can unravel that this solution is

in fact identical, however, denoted in very much less elegant & transparent

way.

This may cause some serious troubles to external programs exploiting Maxima's

functionality. For instance, I discovered this issue while using Sage's `solve`

command -- original bug can be viewed at

https://groups.google.com/forum/?hl=en&fromgroups#!topic/sage-support/ys3CASZ3vrs

Similarly, `solve` also is not capable of recognizing `sin/cos` form of `tan`::

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(%i24) solve(tan(x) = 1, x);

solve: using arc-trig functions to get a solution.

Some solutions will be lost.

                                        %pi

(%o24)                             [x = ---]

                                         4

(%i25) solve((sin(x)/cos(x)) - 1 = 0, x);

(%o25)                         [sin(x) = cos(x)]

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

Is it somehow possible to alter the output for equivalent forms of such

well-known goniometric functions and thus fix this pseudo-bug?

---

Duc Trung Ha

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