It's a little easier to prove 'Sin(a-b)=sinacosb - cosasinb'
(and then change the sign on b).
The basic idea is to set up the points whose coordinates are
(cos(a),sin(a)) (i.e. the point a distance a from (0,0) measured along the circle) and (cos(b),sin(b)) and calculate the straight line distance between them (the arc distance, along the circle, is a-b, of course.) Now mark the point whose arc length from (1,0) is also a-b: it's coordinates are (cos(a-b), sin(a-b)) and calculate the straight line distance beween it and (1,0). Since the arclengths are the same, the lengths of these chords are the same. Set the two calculations equal and "grind".