Simplifying with imaginary numbers

How would I solve 2i/(1+i) ?
 I know the answer is 1+i but I don't know how to get there.

2i
------
1 + i

2i(1 - i)
----------------- {multiplied top and bottom by conjugate of denominator}
(1 + i)(1 - i)

2i - 2i²
----------------- {used distributive property on top and foil method on bottom}
1 - i²

2i - 2(-1)
------------------ {in imaginary numbers, i² is -1}
1 - (-1)

2i + 2
--------- {simplified on top and bottom}
1 + 1

2(i + 1)
------------ {factored 2 out of top}
2

= i + 1 {cancelled the 2's}

= 1 + i {flipped around}

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