The argument of the complex number `((3+i)/(2-i)+(3-i)/(2+i))` is equal to

Options

A. A. `(pi)/2`
B. B. `(pi)/4`
C. C. `0`
D. D. `-(pi)/4`

Answer

To find the argument of the complex number \(\frac{3+i}{2-i} + \frac{3-i}{2+i}\), we will simplify the expression step by step.

### Step 1: Simplify Each Fraction
We start with the expression:
\[ \frac{3+i}{2-i} + \frac{3-i}{2+i} \]

To simplify each fraction, we will multiply the numerator and the denominator by the conjugate of the denominator.

**For the first fraction:**
\[ \frac{3+i}{2-i} \cdot \frac{2+i}{2+i} = \frac{(3+i)(2+i)}{(2-i)(2+i)} \]

Calculating the denominator:
\[ (2-i)(2+i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

Calculating the numerator:
\[ (3+i)(2+i) = 6 + 3i + 2i + i^2 = 6 + 5i - 1 = 5 + 5i \]

Thus, the first fraction simplifies to:
\[ \frac{5 + 5i}{5} = 1 + i \]

**For the second fraction:**
\[ \frac{3-i}{2+i} \cdot \frac{2-i}{2-i} = \frac{(3-i)(2-i)}{(2+i)(2-i)} \]

Calculating the denominator (which is the same as before):
\[ (2+i)(2-i) = 5 \]

Calculating the numerator:
\[ (3-i)(2-i) = 6 - 3i - 2i + i^2 = 6 - 5i - 1 = 5 - 5i \]

Thus, the second fraction simplifies to:
\[ \frac{5 - 5i}{5} = 1 - i \]

### Step 2: Combine the Simplified Fractions
Now we combine both simplified fractions:
\[ (1 + i) + (1 - i) = 1 + i + 1 - i = 2 \]

### Step 3: Find the Argument
The result is a real number \(2\). The argument of a complex number \(z = x + yi\) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]

In our case:
- Real part \(x = 2\)
- Imaginary part \(y = 0\)

Thus, the argument is:
\[ \text{arg}(2) = \tan^{-1}\left(\frac{0}{2}\right) = \tan^{-1}(0) = 0 \]

### Final Answer
The argument of the complex number \(\frac{3+i}{2-i} + \frac{3-i}{2+i}\) is:
\[ \boxed{0} \]