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

Answer

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

### Step 1: Rearranging the equation
We start with the equation:
\[ \cos \theta + \sin \theta = \sqrt{2} \cos \theta \]
Subtract \( \cos \theta \) from both sides:
\[ \sin \theta = \sqrt{2} \cos \theta - \cos \theta \]
This simplifies to:
\[ \sin \theta = (\sqrt{2} - 1) \cos \theta \]

### Step 2: Expressing \(\tan \theta\)
Now, we can divide both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \frac{\sin \theta}{\cos \theta} = \sqrt{2} - 1 \]
This gives us:
\[ \tan \theta = \sqrt{2} - 1 \]

### Step 3: Finding \(\frac{\cos \theta - \sin \theta}{\sin \theta}\)
We need to find:
\[ \frac{\cos \theta - \sin \theta}{\sin \theta} \]
We can rewrite this as:
\[ \frac{\cos \theta}{\sin \theta} - 1 = \cot \theta - 1 \]

### Step 4: Expressing \(\cot \theta\)
Since \( \tan \theta = \sqrt{2} - 1 \), we can find \( \cot \theta \) as:
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{2} - 1} \]

### Step 5: Rationalizing \(\cot \theta\)
To rationalize \( \cot \theta \):
\[ \cot \theta = \frac{1}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{2} + 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} + 1}{2 - 1} = \sqrt{2} + 1 \]

### Step 6: Final calculation
Now substituting back into our expression:
\[ \frac{\cos \theta - \sin \theta}{\sin \theta} = \cot \theta - 1 = (\sqrt{2} + 1) - 1 = \sqrt{2} \]

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