If n is any integer ,then the general solution of the quation `costheta-sintheta=(1)/(sqrt(2))` is

Options

A. `theta=2npi-(pi)/(12)ortheta=2npi+(7pi)/(12)`
B. `theta=npi+(pi)/(12)ortheta=2npi+(7pi)/(12)`
C. `theta=npi+(pi)/(12)`
D. `theta=2npi+(pi)/(12)ortheta=2npi+(7pi)/(12)`

Explanation

To solve the equation \( \cos \theta - \sin \theta = \frac{1}{\sqrt{2}} \), we will follow these steps:

### Step 1: Multiply both sides by \( \frac{1}{\sqrt{2}} \)

We start with the equation:
\[ \cos \theta - \sin \theta = \frac{1}{\sqrt{2}} \]

Multiply both sides by \( \frac{1}{\sqrt{2}} \):
\[ \frac{1}{\sqrt{2}} \cos \theta - \frac{1}{\sqrt{2}} \sin \theta = \frac{1}{2} \]

### Step 2: Rewrite using trigonometric identities

We know that \( \frac{1}{\sqrt{2}} = \cos \frac{\pi}{4} = \sin \frac{\pi}{4} \). Thus, we can rewrite the left side:
\[ \cos \frac{\pi}{4} \cos \theta - \sin \frac{\pi}{4} \sin \theta = \frac{1}{2} \]

This can be expressed using the cosine of a sum formula:
\[ \cos \left( \theta + \frac{\pi}{4} \right) = \frac{1}{2} \]

### Step 3: Solve for \( \theta + \frac{\pi}{4} \)

The general solution for \( \cos x = \frac{1}{2} \) is:
\[ x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \]

Substituting \( x = \theta + \frac{\pi}{4} \):
\[ \theta + \frac{\pi}{4} = 2n\pi \pm \frac{\pi}{3} \]

### Step 4: Isolate \( \theta \)

Now, we isolate \( \theta \):
\[ \theta = 2n\pi \pm \frac{\pi}{3} - \frac{\pi}{4} \]

### Step 5: Simplify the expression

To combine the fractions, we need a common denominator. The least common multiple of 3 and 4 is 12:
\[ \theta = 2n\pi \pm \left( \frac{4\pi}{12} \pm \frac{3\pi}{12} \right) \]
\[ \theta = 2n\pi \pm \frac{7\pi}{12} \quad \text{or} \quad 2n\pi \pm \frac{\pi}{12} \]

### Final General Solution

Thus, the general solutions for \( \theta \) are:
\[ \theta = 2n\pi + \frac{7\pi}{12} \quad \text{or} \quad \theta = 2n\pi - \frac{7\pi}{12} \]