If `F(x)=[(cos^(2)x,cosxsinx),(cosxsinx,sin^(2)x)]` and the difference of `x` and `y` is the odd Multiple of `(pi)/(2),`then `F(x)F(y) `is :

Options

A. Zero matrix
B. unit matrix
C. diagonal matrix
D. None of these

Explanation

To solve the problem, we need to find the product of the matrices \( F(x) \) and \( F(y) \) given that the difference \( x - y \) is an odd multiple of \( \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Define the matrices**:
\[ F(x) = \begin{pmatrix} \cos^2 x & \cos x \sin x \\ \cos x \sin x & \sin^2 x \end{pmatrix} \]
\[ F(y) = \begin{pmatrix} \cos^2 y & \cos y \sin y \\ \cos y \sin y & \sin^2 y \end{pmatrix} \]

2. **Multiply the matrices**:
We need to compute \( F(x) F(y) \):
\[ F(x) F(y) = \begin{pmatrix} \cos^2 x & \cos x \sin x \\ \cos x \sin x & \sin^2 x \end{pmatrix} \begin{pmatrix} \cos^2 y & \cos y \sin y \\ \cos y \sin y & \sin^2 y \end{pmatrix} \]

The product of two matrices is calculated as follows:
\[ F(x) F(y) = \begin{pmatrix} \cos^2 x \cdot \cos^2 y + \cos x \sin x \cdot \cos y \sin y & \cos^2 x \cdot \cos y \sin y + \cos x \sin x \cdot \sin^2 y \\ \cos x \sin x \cdot \cos^2 y + \sin^2 x \cdot \cos y \sin y & \cos x \sin x \cdot \cos y \sin y + \sin^2 x \cdot \sin^2 y \end{pmatrix} \]

3. **Simplify the elements**:
- The first element:
\[ \cos^2 x \cdot \cos^2 y + \cos x \sin x \cdot \cos y \sin y = \cos^2 x \cos^2 y + \frac{1}{2} \sin(2x) \sin(2y) \]
- The second element:
\[ \cos^2 x \cdot \cos y \sin y + \cos x \sin x \cdot \sin^2 y = \cos^2 x \cdot \cos y \sin y + \frac{1}{2} \sin(2x) \sin(2y) \]
- The third element:
\[ \cos x \sin x \cdot \cos^2 y + \sin^2 x \cdot \cos y \sin y = \cos x \sin x \cdot \cos^2 y + \frac{1}{2} \sin(2x) \sin(2y) \]
- The fourth element:
\[ \cos x \sin x \cdot \cos y \sin y + \sin^2 x \cdot \sin^2 y = \frac{1}{2} \sin(2x) \sin(2y) + \sin^2 x \sin^2 y \]

4. **Use the condition \( x - y = (2n + 1) \frac{\pi}{2} \)**:
Since \( x - y \) is an odd multiple of \( \frac{\pi}{2} \), we have:
\[ \cos(x - y) = 0 \]
Thus, every term involving \( \cos(x - y) \) will vanish.

5. **Final result**:
Since \( \cos(x - y) = 0 \), all elements of the resulting matrix will be zero:
\[ F(x) F(y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Conclusion:
The product \( F(x) F(y) \) is the zero matrix.