That is quite interesting:

In[1]:= dot = 
 Integrate[Sin[j x] Sin[k x], {x, 0, 2 Pi}, 
  Assumptions -> j \[Element] Integers \[And] k \[Element] Integers]

Out[1]= 1/2 (Sin[2 (j - k) \[Pi]]/(j - k) - Sin[2 (j + k) \[Pi]]/(
   j + k))

In[2]:= Limit[dot, j -> k, Assumptions -> k \[Element] Integers]

Out[2]= \[Pi]

Daniel,

thanks, this is interesting. Looking at the various discussions I have now seen, it seems that a lot (perhaps all) of the issues center around the way Mathematica's Simplify function works. Now, clearly this is a core, and probably quite complex function, but nevertheless, if my hunch is correct then one could consider modifying the way this function works along the lines I suggested above (a global option to modify simplification behavior, or an option that will have Simplify issue Warning messages when potential solutions get dropped). This way, a user may choose to have Mathematica use a "less aggressive" mode of simplification, that is less likely, or ideally cannot, generate the wrong results we are discussing here.

I think this is a good point: Having an option (perhaps something like "RigorousMath") could be useful. Another possibility would be to generate a warning message, similar to the ones we get for, e.g., solutions of nonlinear equations, where Mathematica does warn us that some solutions may be missing. In both cases the user can then use his/her additional insight into the problem of interest to reformulate the problem, or provide restrictions such that the problem of a ballooning tree of conditionals can be avoided. Let's not forget that there may be cases where the user actually does want to see all of the possible solutions, too. Like you said, in the simple case of Solve versus Reduce, that option exists, but in many other cases it doesn't.

Finally, admittedly on a slightly different note, I would be curious as to whether there is in fact a rigorous definition, that is grounded in mathematical reasoning rather than convenience of coding or using Mathematica, behind that comforting-sounding adjective in "generic solution". Frankly, the term "generic" is a dangerously over-used term in certain circles, often used to gloss over significant holes in people's mathematical arguments...