if |z|=1 proove z-1/z+1 is purely imazinary

Question

anonymous · Answer

@myko can you do this one? I don't know how to go about it.

anonymous · Answer

sorryy i donot know thats why i m askng

anonymous · Answer

would making z-1/z+1 = (z+1-2)/z+1 do anything productive?

anonymous · Answer

noo noooo we just have to prove this

anonymous · Answer

then we could make it 1-(2/z+1)...

anonymous · Answer

ok, lets take z= x+iy, then
$\frac{x+iy-1}{x+iy+1}=\frac{x-1}{(x+1)+iy}+\frac{iy}{(x+1)+iy}=\frac{(x-1)[(x+1)-iy]+iy[(x+1)-iy]}{(x+1)^2+y^2}=\frac{x^2+y^2-1+i[(x-1)(x+1)]}{(x+1)^2+y^2}$
now since |z|=1, it means $x^2+y^2=1$ so you will be left with juast purly imaginary number

anonymous · Answer

@Peter14

anonymous · Answer

looks right
I trust people to multiply right, and it looks like that works.

anonymous · Answer

just a litle typo, in the last fraction switch places (x-1) and x+1) and put - sign in between them