Solveset fails with trig and symbols

[oscarbenjamin]

Failing example:

In [2]: solveset(cos(x)-y, x, Interval(0, pi))                                                                                                 
Out[2]: {x | x ∊ ℝ ∧ -y + cos(x) = 0}

Firstly the domain of the ConditionSet is wrong. It should be Interval(0, pi) not Reals.

Secondly cos is invertible over that domain so we should be able to get acos(y). We don't know that y is real and between -1 and 1 but a better answer could be

In [12]: FiniteSet(acos(y)) & Interval(0, pi)                                                                                                  
Out[12]: [0, π] ∩ {acos(y)}

[oscarbenjamin]

Also solveset returns very complicated answers with tan and irrational numbers:

In [17]: solveset(tan(x)-S.Half, x, Interval(0, pi/2))       
Out[17]: {atan(1/2)}

In [18]: solveset(tan(x)-sqrt(2), x, Interval(0, pi/2))                                                                                        
Out[18]: 
⎧   ⎛                                                       ⎛   ⎛atan(2⋅√2)⎞⎞⎞⎫
⎪   ⎜   ⎛    _____________________________________⎞         ⎜cos⎜──────────⎟⎟⎟⎪
⎪   ⎜   ⎜   ╱    2⎛atan(2⋅√2)⎞      2⎛atan(2⋅√2)⎞ ⎟         ⎜   ⎝    2     ⎠⎟⎟⎪
⎨-ⅈ⋅⎜log⎜  ╱  sin ⎜──────────⎟ + cos ⎜──────────⎟ ⎟ + ⅈ⋅atan⎜───────────────⎟⎟⎬
⎪   ⎜   ⎝╲╱       ⎝    2     ⎠       ⎝    2     ⎠ ⎠         ⎜   ⎛atan(2⋅√2)⎞⎟⎟⎪
⎪   ⎜                                                       ⎜sin⎜──────────⎟⎟⎟⎪
⎩   ⎝                                                       ⎝   ⎝    2     ⎠⎠⎠⎭

In [19]: solveset(tan(x)-pi, x, Interval(0, pi/2))                                                                                             
Out[19]: 
⎧   ⎛   ⎛            ________________________________________________________________________________________________________________________
⎪   ⎜   ⎜           ╱                                                ⎛    ⎛       2⋅π       ⎞⎞                                               
⎪   ⎜   ⎜          ╱                                                 ⎜atan⎜─────────────────⎟⎟                                               
⎪   ⎜   ⎜         ╱                                                  ⎜    ⎜              4  ⎟⎟                                               
⎪   ⎜   ⎜        ╱            __________________________________     ⎜    ⎜    1        π   ⎟⎟           __________________________________  
⎪   ⎜   ⎜       ╱            ╱                                2      ⎜    ⎜- ────── + ──────⎟⎟          ╱                                2   
⎪   ⎜   ⎜      ╱            ╱        2     ⎛              2  ⎞       ⎜    ⎜       2        2⎟⎟         ╱        2     ⎛              2  ⎞    
⎨   ⎜   ⎜     ╱            ╱      4⋅π      ⎜    1        π   ⎟      2⎜    ⎝  1 + π    1 + π ⎠⎟        ╱      4⋅π      ⎜    1        π   ⎟    
⎪-ⅈ⋅⎜log⎜    ╱            ╱    ───────── + ⎜- ────── + ──────⎟  ⋅sin ⎜───────────────────────⎟ +     ╱    ───────── + ⎜- ────── + ──────⎟  ⋅c
⎪   ⎜   ⎜   ╱            ╱             2   ⎜       2        2⎟       ⎝           2           ⎠      ╱             2   ⎜       2        2⎟    
⎪   ⎜   ⎜  ╱            ╱      ⎛     2⎞    ⎝  1 + π    1 + π ⎠                                     ╱      ⎛     2⎞    ⎝  1 + π    1 + π ⎠    
⎪   ⎜   ⎝╲╱           ╲╱       ⎝1 + π ⎠                                                          ╲╱       ⎝1 + π ⎠                           
⎪   ⎜                                                                                                                                        
⎪   ⎜                                                                                                                                        
⎩   ⎝                                                                                                                                        

_____________________________⎞                            ⎞⎫
   ⎛    ⎛       2⋅π       ⎞⎞ ⎟                            ⎟⎪
   ⎜atan⎜─────────────────⎟⎟ ⎟         ⎛   ⎛    ⎛ 2⋅π  ⎞⎞⎞⎟⎪
   ⎜    ⎜              4  ⎟⎟ ⎟         ⎜   ⎜atan⎜──────⎟⎟⎟⎟⎪
   ⎜    ⎜    1        π   ⎟⎟ ⎟         ⎜   ⎜    ⎜     2⎟⎟⎟⎟⎪
   ⎜    ⎜- ────── + ──────⎟⎟ ⎟         ⎜   ⎜    ⎝1 - π ⎠⎟⎟⎟⎪
   ⎜    ⎜       2        2⎟⎟ ⎟         ⎜cos⎜────────────⎟⎟⎟⎪
  2⎜    ⎝  1 + π    1 + π ⎠⎟ ⎟         ⎜   ⎝     2      ⎠⎟⎟⎬
os ⎜───────────────────────⎟ ⎟ - ⅈ⋅atan⎜─────────────────⎟⎟⎪
   ⎝           2           ⎠ ⎟         ⎜   ⎛    ⎛ 2⋅π  ⎞⎞⎟⎟⎪
                             ⎟         ⎜   ⎜atan⎜──────⎟⎟⎟⎟⎪
                             ⎠         ⎜   ⎜    ⎜     2⎟⎟⎟⎟⎪
                                       ⎜   ⎜    ⎝1 - π ⎠⎟⎟⎟⎪
                                       ⎜sin⎜────────────⎟⎟⎟⎪
                                       ⎝   ⎝     2      ⎠⎠⎠⎭

[gschintgen]

Also solveset returns very complicated answers with tan and irrational numbers:

That's because equations of purely trigonometric nature end up in solve_trig which currently doesn't call _invert. See also: #17334