Find the general solution of `Sin x-Cos x = 1`.

Answer

To find the general solution of the equation \( \sin x - \cos x = 1 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin x - \cos x = 1 \]

### Step 2: Divide by \( \sqrt{2} \)
To simplify the equation, we divide both sides by \( \sqrt{2} \):
\[ \frac{\sin x}{\sqrt{2}} - \frac{\cos x}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 3: Recognize trigonometric identities
We know that:
\[ \frac{1}{\sqrt{2}} = \cos\left(\frac{\pi}{4}\right) \quad \text{and} \quad \frac{1}{\sqrt{2}} = \sin\left(\frac{\pi}{4}\right) \]
Thus, we can rewrite the equation as:
\[ \sin x \cdot \sin\left(\frac{\pi}{4}\right) - \cos x \cdot \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 4: Use the sine subtraction formula
Using the sine subtraction formula \( \sin(a - b) = \sin a \cos b - \cos a \sin b \), we can express the left-hand side:
\[ \sin\left(x - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 5: Solve for \( x - \frac{\pi}{4} \)
We know that \( \sin y = \frac{1}{\sqrt{2}} \) corresponds to angles:
\[ y = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad y = \frac{3\pi}{4} + 2n\pi \]
where \( n \) is any integer.

### Step 6: Substitute back for \( x \)
Now, substituting back for \( y \):
1. From \( x - \frac{\pi}{4} = \frac{\pi}{4} + 2n\pi \):
\[ x = \frac{\pi}{4} + \frac{\pi}{4} + 2n\pi = \frac{\pi}{2} + 2n\pi \]
2. From \( x - \frac{\pi}{4} = \frac{3\pi}{4} + 2n\pi \):
\[ x = \frac{3\pi}{4} + \frac{\pi}{4} + 2n\pi = \pi + 2n\pi \]

### Step 7: Combine the solutions
The general solution can be expressed as:
\[ x = \frac{\pi}{2} + 2n\pi \quad \text{or} \quad x = \pi + 2n\pi \]

### Final General Solution
Thus, the complete general solution is:
\[ x = \frac{\pi}{2} + 2n\pi \quad \text{and} \quad x = \pi + 2n\pi \quad \text{for } n \in \mathbb{Z} \]