If `((1+i)/(1 -i)) - ((1 - i)/(1 +i))` = x +iy find x and y .

Answer

To solve the equation \(\frac{1+i}{1-i} - \frac{1-i}{1+i} = x + iy\), we can follow these steps:

### Step 1: Simplify the first term
We start with the first term \(\frac{1+i}{1-i}\). To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{1+i}{1-i} \cdot \frac{1+i}{1+i} = \frac{(1+i)(1+i)}{(1-i)(1+i)} \]

Calculating the numerator:

\[ (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \]

Calculating the denominator:

\[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 2 \]

So, we have:

\[ \frac{1+i}{1-i} = \frac{2i}{2} = i \]

### Step 2: Simplify the second term
Now we simplify the second term \(\frac{1-i}{1+i}\) in a similar manner:

\[ \frac{1-i}{1+i} \cdot \frac{1-i}{1-i} = \frac{(1-i)(1-i)}{(1+i)(1-i)} \]

Calculating the numerator:

\[ (1-i)(1-i) = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \]

Calculating the denominator:

\[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2 \]

So, we have:

\[ \frac{1-i}{1+i} = \frac{-2i}{2} = -i \]

### Step 3: Combine the two simplified terms
Now we can substitute back into the original equation:

\[ \frac{1+i}{1-i} - \frac{1-i}{1+i} = i - (-i) = i + i = 2i \]

### Step 4: Equate to \(x + iy\)
Now we can equate this to \(x + iy\):

\[ 2i = 0 + 2i \]

From this, we can see that:

\[ x = 0 \quad \text{and} \quad y = 2 \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:

\[ x = 0, \quad y = 2 \]

---