That depends on exactly how you define the trig functions themselves. One common method is this: Give a non-negative number t, start at the point (1, 0) and measure counterclockwise around the circumference a distance t. The final point will have coordinates (cos(t), sin(t)). In other words, cos(t) is defined as the x-coordinate of that point and sin(t) is defined as the y-coordinate. Given that, consider the straight line distance between (cos(a+b), sin(a+b)) and (cos(a),sin(a)) and the straight line distance between (0, 0) and (cos(b), sin(b)). The arc between the first pair of points has length b and the arc between the second part also has length b so the chords (straight lines between them) must also have the same length.

Squaring both sides of that gives

so

Now, that isn't quite what we want but if we take x= a, y= a+b, we can "reverse" that: now b= y- a= y- x so that becomes

Replacing x by -x and using the fact that cosine is an even function and sine is an odd function.

sin(y+x) can be done now by using the fact that and .