[math-fun] Does this define sin(z)?

Wouldn't a constant sin(z) = 0 satisfy these conditions? Tom Bill Gosper writes:

1) sin(0) := 0 and

2) |sin(x) - sin(y)| ≤ |x - y| and

3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2).

Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg