I guess you mean
(1 + i ) / (1 - i ) + 1 / (1 + i )
in which case, notice that the denominator of both fractions are conjugates of one another. Combine the fractions by multiplying either fraction by the appropriate conjugate:
[(1 + i ) / (1 - i ) • (1 + i ) / (1 + i )]+ [1 / (1 + i ) • (1 - i ) / (1 - i )]
In either denominator, we have a difference of squares:
(1 + i ) (1 - i ) = 1² + i - i - i ² = 1² - i ² = 1 + 1 = 2
→ (1 + i )² / 2 + (1 - i ) / 2
→ ((1 + i )² + (1 - i )) / 2
Expand the numerator:
(1 + i )² + (1 - i ) = (1² + 2i + i ²) + 1 - i = 1 + 2 - 1 + 1 - i = 3 - i
and so
(1 + i ) / (1 - i ) + 1 / (1 + i ) = (3 - i ) / 2
