Counting problem with Mobieus function

[soopo]

Homework Statement

How can you get from this
$$\frac {z(i-1) +i +1} {z(1-i) +i +1}$$
to this
$$= \frac { z-1 } {-z -i}$$
?

The Attempt at a Solution

SageMath does not simplify the result any further from the beginning.
The equivalence is based on some high Math.

I am not sure how you can deduce the equivalence.

 

[csco]

The expressions are not equal. Try z = 0, you get 1 in the first expression and -i in the second.
The valid equation is:

$$\frac {z-i} {-z-i} = \frac {z(i-1) +i +1} {z(1-i) +i +1}$$

To check that just multiply and divide the LHS by (i - 1)