$ ext{If } \cos heta + \sin heta = \sqrt{2} \cos heta ext{, then the value of } \cos heta - \sin heta ext{ is :}$

$\text{If } \cos \theta + \sin \theta = \sqrt{2} \cos \theta \text{, then the value of } \cos \theta - \sin \theta \text{ is :}$

Question

$	ext{If } \cos 	heta + \sin 	heta = \sqrt{2} \cos 	heta 	ext{, then the value of } \cos 	heta - \sin 	heta 	ext{ is :}$

Accepted Answer

To solve the problem and justify why Option 2 is correct, we start with the given equation:

$$
\cos \theta + \sin \theta = \sqrt{2} \cos \theta
$$

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

### Step-by-step Explanation:

1. **Rearrange the Given Equation:**

$$
   \cos \theta + \sin \theta = \sqrt{2} \cos \theta
   $$

Subtract $\cos \theta$ from both sides:

$$
   \sin \theta = \sqrt{2} \cos \theta - \cos \theta
   $$

Factor out $\cos \theta$ on the right side:

$$
   \sin \theta = \cos \theta (\sqrt{2} - 1)
   $$

2. **Express $\sin \theta$ in terms of $\cos \theta$:**

Divide both sides by $\cos \theta$ (assuming $\cos \theta \neq 0$):

$$
   \tan \theta = \sqrt{2} - 1
   $$

3. **Find $\cos \theta - \sin \theta$:**

We know:

$$
   \cos \theta - \sin \theta = \cos \theta - \cos \theta (\sqrt{2} - 1)
   $$

Simplify the expression:

$$
   \cos \theta - \sin \theta = \cos \theta (1 - (\sqrt{2} - 1))
   $$

$$
   \cos \theta - \sin \theta = \cos \theta (2 - \sqrt{2})
   $$

4. **Relate to the Given Options:**

We need to express $\cos \theta - \sin \theta$ in terms of $\sin \theta$. From the earlier step, we have:

$$
   \sin \theta = \cos \theta (\sqrt{2} - 1)
   $$

Therefore, $\cos \theta = \frac{\sin \theta}{\sqrt{2} - 1}$.

Substitute back:

$$
   \cos \theta - \sin \theta = \frac{\sin \theta}{\sqrt{2} - 1} (2 - \sqrt{2})
   $$

Simplifying further, we find:

$$
   \cos \theta - \sin \theta = \sqrt{2} \sin \theta
   $$

This matches Option 2.

### Explanation of Incorrect Options:

1. **Option 1: $\sqrt{2} \tan \theta$**

This option suggests a relationship involving $\tan \theta$, which does not directly relate to the expression $\cos \theta - \sin \theta$ as derived.

2. **Option 3: $\sqrt{2} \cot \theta$**

This option involves $\cot \theta$, which is not relevant to the expression derived for $\cos \theta - \sin \theta$.

3. **Option 4: $\sqrt{2} \text{cosec} \theta$**

This option involves $\text{cosec} \theta$, which does not align with the expression derived for $\cos \theta - \sin \theta$.

Thus, the correct answer is indeed Option 2: $\sqrt{2} \sin \theta$.