sin\frac{1}{z-1}= sin[(\frac{1}{z-1}-1)+1]=sin(\frac{1}{z-1}-1)*cos1 + cos(\frac{1}{z-1})*sin1= cos1*\sum_{n=0}^\infty \frac{(-1)^n(\frac{1}{z-1}-1)^{2n+1} }{(2n+1)!} + sin1*\sum_{n=0}^\infty \frac{(-1)^n (\frac{1}{z-1}-1)^{2n} }{(2n)!}