`(( 2 i)/( 1 + i))^(2) =`

Options

A. i
B. 2i
C. 1-i
D. 1-2i

Answer

To solve the expression \(\left(\frac{2i}{1+i}\right)^2\), we will follow these steps:

### Step 1: Square the Numerator
First, we square the numerator \(2i\):
\[ (2i)^2 = 4i^2 \]
Since \(i^2 = -1\), we have:
\[ 4i^2 = 4(-1) = -4 \]

### Step 2: Square the Denominator
Next, we need to square the denominator \(1+i\):
\[ (1+i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i + (-1) = 2i \]

### Step 3: Combine the Results
Now we can combine the results from Steps 1 and 2:
\[ \left(\frac{2i}{1+i}\right)^2 = \frac{-4}{2i} \]

### Step 4: Simplify the Expression
We simplify \(\frac{-4}{2i}\):
\[ \frac{-4}{2i} = \frac{-2}{i} \]

### Step 5: Multiply by the Conjugate
To eliminate \(i\) from the denominator, we multiply the numerator and denominator by \(i\):
\[ \frac{-2}{i} \cdot \frac{i}{i} = \frac{-2i}{i^2} \]
Since \(i^2 = -1\):
\[ \frac{-2i}{-1} = 2i \]

### Final Result
Thus, the final result is:
\[ \left(\frac{2i}{1+i}\right)^2 = 2i \]

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