Question

If (1+i)z =(1-i)z conjugated show z =-iz conjugated

Answer 1

( 1 + i)z = (1 - i)z'

Now,

To prove ,

z= -z'

Proof:

So,

1 + i in polar form can be written as,

√2 (e^(i×pi) / 2)

And,

1- i = √2(e^(-i×pi) / 2)

So,

( 1 +i)z = (1 - i)z'

√2 (e^(i×pi) / 2) z = √2(e^(-i×pi) / 2)z'

z= (e^(-i×pi/2 - i× pi/ 2))z'

z= (e^(-i×pi)) z'

Now,

(e^(-i×pi) = cos (pi) + isin (pi)

= -1

So,

z= -z'

Answer 2

In the attachment I have answered this problem.

The given expression is simplified in such a way to get the required result.

The conjugate of 1+i is 1-i

Conjugate is taken in both numerator and denominator . After that simplifications are done to get required result.

See the attachment for detailed solution