If `sintheta+costheta=sqrt2costhetathen costheta-sintheta` is equal to

Options

A. `sqrt2cos theta`
B. `sqrt2 sintheta`
C. `sqrt2(cos theta+sintheta)`
D. none of these

Answer

To solve the problem where \( \sin \theta + \cos \theta = \sqrt{2} \cos \theta \) and we need to find the value of \( \cos \theta - \sin \theta \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \sin \theta + \cos \theta = \sqrt{2} \cos \theta \]

### Step 2: Rearrange the equation
Rearranging gives:
\[ \sin \theta = \sqrt{2} \cos \theta - \cos \theta \]
This simplifies to:
\[ \sin \theta = (\sqrt{2} - 1) \cos \theta \]

### Step 3: Square both sides
Now, square both sides:
\[ \sin^2 \theta = (\sqrt{2} - 1)^2 \cos^2 \theta \]
Expanding the right side:
\[ \sin^2 \theta = (2 - 2\sqrt{2} + 1) \cos^2 \theta = (3 - 2\sqrt{2}) \cos^2 \theta \]

### Step 4: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting this into the equation gives:
\[ 1 - \cos^2 \theta = (3 - 2\sqrt{2}) \cos^2 \theta \]

### Step 5: Combine like terms
Rearranging the equation:
\[ 1 = (3 - 2\sqrt{2} + 1) \cos^2 \theta \]
This simplifies to:
\[ 1 = (4 - 2\sqrt{2}) \cos^2 \theta \]

### Step 6: Solve for \( \cos^2 \theta \)
Now, solving for \( \cos^2 \theta \):
\[ \cos^2 \theta = \frac{1}{4 - 2\sqrt{2}} \]

### Step 7: Find \( \sin^2 \theta \)
Using \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \frac{1}{4 - 2\sqrt{2}} = \frac{(4 - 2\sqrt{2}) - 1}{4 - 2\sqrt{2}} = \frac{3 - 2\sqrt{2}}{4 - 2\sqrt{2}} \]

### Step 8: Find \( \cos \theta - \sin \theta \)
Now, we can find \( \cos \theta - \sin \theta \):
Using the identity:
\[ (\cos \theta - \sin \theta)^2 = \cos^2 \theta + \sin^2 \theta - 2\sin \theta \cos \theta \]
We already know \( \cos^2 \theta + \sin^2 \theta = 1 \), and we can find \( \sin \theta \cos \theta \) using the earlier equations.

### Final Result
After simplifying, we find:
\[ \cos \theta - \sin \theta = \sqrt{2} \sin \theta \]