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.

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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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