If cos x sin x = 0.22, then `(cos x - sin x)^(2)=`

Options

A. 0
B. `0.11`
C. `0.44`
D. `0.56`

Explanation

To solve the problem, we need to find the value of \((\cos x - \sin x)^2\) given that \(\cos x \sin x = 0.22\).

### Step-by-Step Solution:

1. **Start with the identity for the square of a difference:**
\[ (\cos x - \sin x)^2 = \cos^2 x - 2 \cos x \sin x + \sin^2 x \]

2. **Use the Pythagorean identity:**
We know that:
\[ \cos^2 x + \sin^2 x = 1 \]
Therefore, we can substitute this into our equation:
\[ (\cos x - \sin x)^2 = 1 - 2 \cos x \sin x \]

3. **Substitute the given value:**
We are given that \(\cos x \sin x = 0.22\). Substitute this value into the equation:
\[ (\cos x - \sin x)^2 = 1 - 2 \times 0.22 \]

4. **Calculate \(2 \times 0.22\):**
\[ 2 \times 0.22 = 0.44 \]

5. **Subtract from 1:**
\[ (\cos x - \sin x)^2 = 1 - 0.44 = 0.56 \]

### Final Answer:
\[ (\cos x - \sin x)^2 = 0.56 \]