if z = a + bi and|z| = 1, let w = (z+1)/(z-1). prove that w is a pure imaginary number of the form bi.

so from the given i have a^2 + b^2 = 1 since sqrt(a^2 + b^2) = 1 = |z|.

so i plugged in a + bi for z into w = ((a + bi) + 1) / ((a + bi) - 1) i multiplied the top and bottom by the conjugate of the bottom witch is ((a + bi) + 1) and i got something complicated: (2a^2 + 2abi + 2a + 2bi) / (2abi - 2b^2) when i split this up into separate fractions i don't get a pure imaginary number. how do i do this?