an excellent question !

i think this approach MAY be useful!!
break the summation into sin x+y + sin x-y
then apply complex number(substitution of each of the sin function in terms of e)
e^i theta - e^-i theta/2i
and then proceed!!
may be a long confusing process.....
crrct me if i m wrong!!!!!!!!

i mean
it is

now it is
method of difference