If `costheta-sintheta=sqrt(2)costheta`, then find the value of `costheta+sintheta`.

Options

A. `sqrt(2)costheta`
B. `sqrt(2)sintheta`
C. `2sintheta`
D. `sqrt(2)tantheta`

Answer

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

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

### Step 2: Divide both sides by \( \cos \theta \)
Assuming \( \cos \theta \neq 0 \), we can divide both sides by \( \cos \theta \):
\[ 1 - \sqrt{2} = \tan \theta \]

### Step 3: Use the identity for \( \cos \theta + \sin \theta \)
We know that:
\[ \cos \theta + \sin \theta = \sqrt{(\cos \theta)^2 + (\sin \theta)^2 + 2 \cos \theta \sin \theta} \]
Using the Pythagorean identity, \( \cos^2 \theta + \sin^2 \theta = 1 \), we can express:
\[ \cos \theta + \sin \theta = \sqrt{1 + 2 \cos \theta \sin \theta} \]

### Step 4: Find \( \cos \theta \sin \theta \)
From the previous step, we have:
\[ \sin \theta = (1 - \sqrt{2}) \cos \theta \]
Thus:
\[ \cos \theta \sin \theta = \cos \theta \cdot (1 - \sqrt{2}) \cos \theta = (1 - \sqrt{2}) \cos^2 \theta \]

### Step 5: Substitute back into the expression
Now substitute \( \cos \theta \sin \theta \) back into the expression for \( \cos \theta + \sin \theta \):
\[ \cos \theta + \sin \theta = \sqrt{1 + 2(1 - \sqrt{2}) \cos^2 \theta} \]

### Step 6: Solve for \( \cos \theta + \sin \theta \)
To find the exact value, we can use the earlier derived equation. We can square both sides of the equation \( (1 - \sqrt{2}) \cos \theta = \sin \theta \) and use the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This leads us to find \( \cos \theta + \sin \theta \) in terms of known quantities.

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

### Conclusion
Thus, the value of \( \cos \theta + \sin \theta \) is \( \sqrt{2} \sin \theta \), which corresponds to option 2.

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