Write the square of `(i)/(1+i)` in the form x+iy.

Answer

To find the square of \(\frac{i}{1+i}\) in the form \(x + iy\), we will follow these steps:

### Step 1: Write the expression for squaring
We start with the expression:
\[ \left(\frac{i}{1+i}\right)^2 \]

### Step 2: Simplify the fraction
First, we simplify \(\frac{i}{1+i}\). We can multiply the numerator and the denominator by the conjugate of the denominator, which is \(1-i\):
\[ \frac{i(1-i)}{(1+i)(1-i)} \]

### Step 3: Calculate the denominator
Now, we calculate the denominator:
\[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 4: Calculate the numerator
Next, we calculate the numerator:
\[ i(1-i) = i - i^2 = i - (-1) = i + 1 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{i + 1}{2} = \frac{1}{2} + \frac{i}{2} \]

### Step 6: Square the simplified expression
Now we need to square this result:
\[ \left(\frac{1}{2} + \frac{i}{2}\right)^2 \]

### Step 7: Apply the formula for squaring a binomial
Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = \frac{1}{2}\) and \(b = \frac{i}{2}\):
\[ \left(\frac{1}{2}\right)^2 + 2\left(\frac{1}{2}\right)\left(\frac{i}{2}\right) + \left(\frac{i}{2}\right)^2 \]

### Step 8: Calculate each term
Calculating each term:
1. \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
2. \(2\left(\frac{1}{2}\right)\left(\frac{i}{2}\right) = \frac{1}{2}i\)
3. \(\left(\frac{i}{2}\right)^2 = \frac{i^2}{4} = \frac{-1}{4}\)

### Step 9: Combine the results
Now, combine the results:
\[ \frac{1}{4} + \frac{-1}{4} + \frac{1}{2}i = 0 + \frac{1}{2}i \]

### Final Result
Thus, the square of \(\frac{i}{1+i}\) in the form \(x + iy\) is:
\[ 0 + \frac{1}{2}i \]

### Summary
In conclusion, we have:
\[ x = 0, \quad y = \frac{1}{2} \]