I think that 0 is the only fixed point of the sin function in the complex unit disc.
In other words sinz=z. I'm not sure how to prove it.
You could also try to use the argument principle to measure the number of roots in the disk, but the function might be a little nasty to deal with. Putting ,
gives the number of roots in the disk. If you can evaluate this, then you are done.
Maybe somebody has a cleaner way of doing this.