`((1+i)/(1-i))^(2) + ((1-i)/(1+i))^(2)` is equal to :

Options

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

Answer

To solve the expression \(\left(\frac{1+i}{1-i}\right)^{2} + \left(\frac{1-i}{1+i}\right)^{2}\), we will 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 the denominator by the conjugate of the denominator, which is \(1+i\):

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

Calculating the numerator:

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

Calculating the denominator:

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

Thus, we have:

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

### Step 2: Square the first term
Now we square the result from Step 1:

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

### Step 3: Simplify the second term
Next, we simplify the second term \(\frac{1-i}{1+i}\) in a similar way. We multiply the numerator and the denominator by the conjugate of the denominator, which is \(1-i\):

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

Calculating the numerator:

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

Calculating the denominator:

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

Thus, we have:

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

### Step 4: Square the second term
Now we square the result from Step 3:

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

### Step 5: Combine the results
Now we combine the results from Step 2 and Step 4:

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

### Final Answer
Thus, the final answer is:

\[ \boxed{-2} \]