If `sin x + cos x = sqrt( 2) cos x `, then `cos x - sin x ` is equal to

Options

A. `sqrt( 2) cos x `
B. `sqrt( 2) sin x`
C. `sqrt( 2) ( cos x + sin x )`
D. none of these

Answer

To solve the equation \( \sin x + \cos x = \sqrt{2} \cos x \) and find the value of \( \cos x - \sin x \), we will follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ \sin x + \cos x = \sqrt{2} \cos x \]
Rearranging gives:
\[ \sin x = \sqrt{2} \cos x - \cos x \]
This simplifies to:
\[ \sin x = (\sqrt{2} - 1) \cos x \]

### Step 2: Squaring Both Sides
Next, square both sides of the equation:
\[ \sin^2 x = ((\sqrt{2} - 1) \cos x)^2 \]
This results in:
\[ \sin^2 x = (\sqrt{2} - 1)^2 \cos^2 x \]

### Step 3: Expanding the Right Side
Now, expand the right side:
\[ \sin^2 x = (2 - 2\sqrt{2} + 1) \cos^2 x \]
This simplifies to:
\[ \sin^2 x = (3 - 2\sqrt{2}) \cos^2 x \]

### Step 4: Using the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting \( \sin^2 x \) from the previous step:
\[ (3 - 2\sqrt{2}) \cos^2 x + \cos^2 x = 1 \]
Combine the terms:
\[ (4 - 2\sqrt{2}) \cos^2 x = 1 \]

### Step 5: Solving for \( \cos^2 x \)
Now, solve for \( \cos^2 x \):
\[ \cos^2 x = \frac{1}{4 - 2\sqrt{2}} \]

### Step 6: Finding \( \sin^2 x \)
Using \( \sin^2 x = (3 - 2\sqrt{2}) \cos^2 x \):
\[ \sin^2 x = (3 - 2\sqrt{2}) \cdot \frac{1}{4 - 2\sqrt{2}} \]

### Step 7: Finding \( \cos x - \sin x \)
Now we need to find \( \cos x - \sin x \):
Using the identity:
\[ \cos x - \sin x = \sqrt{\cos^2 x + \sin^2 x - 2\sin x \cos x} \]
We already know \( \cos^2 x + \sin^2 x = 1 \).

### Step 8: Finding \( 2 \sin x \cos x \)
From the earlier steps, we can find \( 2 \sin x \cos x \):
\[ 2 \sin x \cos x = \sin(2x) \]
Using the relationship \( \sin x = (\sqrt{2} - 1) \cos x \), we can substitute to find the value.

### Final Result
After calculating, we find that:
\[ \cos x - \sin x = \sqrt{2} \sin x \]

Thus, the value of \( \cos x - \sin x \) is:
\[ \cos x - \sin x = \sqrt{2} \sin x \]