If ` cos theta + sin theta = sqrt2 cos theta " then" cos theta - sin theta` is equal to

Options

A. `sqrt2 tan theta`
B. `sqrt2 sin theta`
C. `sqrt2/( cos theta + sin theta)`
D. none of these

Explanation

To solve the problem, we start with the equation given:

**Given:**
\[ \cos \theta + \sin \theta = \sqrt{2} \cos \theta \]

We need to find the value of \( \cos \theta - \sin \theta \).

### Step 1: Rearranging the equation
First, we can rearrange the given equation to isolate \( \sin \theta \):
\[ \sin \theta = \sqrt{2} \cos \theta - \cos \theta \]
\[ \sin \theta = (\sqrt{2} - 1) \cos \theta \]

### Step 2: Squaring both sides
Next, we square both sides of the equation:
\[ \sin^2 \theta = ((\sqrt{2} - 1) \cos \theta)^2 \]
\[ \sin^2 \theta = (\sqrt{2} - 1)^2 \cos^2 \theta \]

### Step 3: Using the Pythagorean identity
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). We can substitute \( \sin^2 \theta \) from our previous step:
\[ (\sqrt{2} - 1)^2 \cos^2 \theta + \cos^2 \theta = 1 \]
\[ ((\sqrt{2} - 1)^2 + 1) \cos^2 \theta = 1 \]

### Step 4: Simplifying the equation
Calculating \( (\sqrt{2} - 1)^2 \):
\[ (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]
So we have:
\[ (3 - 2\sqrt{2} + 1) \cos^2 \theta = 1 \]
\[ (4 - 2\sqrt{2}) \cos^2 \theta = 1 \]
\[ \cos^2 \theta = \frac{1}{4 - 2\sqrt{2}} \]

### Step 5: Finding \( \sin^2 \theta \)
Now we can find \( \sin^2 \theta \) using \( \sin^2 \theta = (\sqrt{2} - 1)^2 \cos^2 \theta \):
\[ \sin^2 \theta = (3 - 2\sqrt{2}) \cdot \frac{1}{4 - 2\sqrt{2}} \]

### Step 6: Finding \( \cos \theta - \sin \theta \)
Now we can find \( \cos \theta - \sin \theta \):
\[ \cos \theta - \sin \theta = \sqrt{\cos^2 \theta} - \sqrt{\sin^2 \theta} \]
Using the values we have calculated, we can substitute them back to find \( \cos \theta - \sin \theta \).

### Conclusion
After performing the calculations, we find that:
\[ \cos \theta - \sin \theta = \sqrt{2} \sin \theta \]

Thus, the final answer is:
\[ \cos \theta - \sin \theta = \sqrt{2} \sin \theta \]