I am not sure that this is what you want, but here it goes...
If you want to find an expression for sin(p)+sin(q) of the form
2*sin(a+b)*cos(a-b), you can write an equation
(%i1) eq:sin(p)+sin(q)=2*sin(a+b)*cos(a-b)$
and then try to solve for a,b. There is a command "solve" for this,
but it is not useful here. What you can do is to express the members of
the equation in canonical form (with "trigrat"), and identify terms:
(%i2) trigrat(eq);
(%o2) sin(q)+sin(p)=sin(2*b)+sin(2*a)
From this equation, you get that a=p/2, b=q/2:
(%i3) factor(subst([a=p/2,b=q/2],rhs(eq)));
(%o3) 2*cos((q-p)/2)*sin((q+p)/2)
If what you want is just to replace sin(p)+sin(q) for
2*sin(a+b)*cos(a-b) in an equation, to solve it, you can do something like
(%i4) tellsimp(sin(p)+sin(q),2*sin(a+b)*cos(a-b))$
which tells the simplifier to replace sin(p)+sin(q) by 2*sin(a+b)*cos(a-b).
I hope this helps,
J. A.