In the enclosed notebook, the functions ts and ts2 are equivalent, both in theory and in Mathematica computation. However, when used in the same formula, see z3b1 and z3b2, the former works, the latter returns complex infinity. Is this a bug or is there some subtlety that I am unaware of? Thanks.
Leon
Gianluca, I see, you are referring to the function ts. That does look like a nice way to prove the result, which is
Sum[Sin[A + i D], {i, 0, n - 1}] = ( Sin[(n D)/2] Sin[A + ((n - 1) D)/2] )/Sin[D/2]
Here's another example where Mathematica fails.
If I am not mistaken, your sum is a geometric progression:
Sin[(i \[Pi])/2^(n - 1) (3/2 + k)] // TrigToExp % /. Exp[a_] :> (Exp[Cancel[a/i]])^i
