lf `sin(x-y)-cos(x + y) = 1/2` then the values of `x & y` lying between `0 and pi` are given by

Options

A. `x=45^(@), y = 15^(@)`
B. `x=45^(@), y=135^(@)`
C. `x=165^(@), y=15^(@)`
D. `x=165^(@), y=135^(@)`

Answer

To solve the equation \( \sin(x - y) - \cos(x + y) = \frac{1}{2} \), we can break it down into two separate equations based on the given trigonometric identities.

### Step 1: Set up the equations
From the equation, we can express it as:
\[ \sin(x - y) = \frac{1}{2} + \cos(x + y) \]

### Step 2: Analyze the cosine term
We know that \( \cos(x + y) \) can take values between -1 and 1. Therefore, we can set up the conditions for \( \sin(x - y) \):
\[ \sin(x - y) = \frac{1}{2} + \cos(x + y) \]
This means that \( \sin(x - y) \) must also lie between 0 and 1, which gives us:
\[ \frac{1}{2} - 1 \leq \sin(x - y) \leq \frac{1}{2} + 1 \]
This simplifies to:
\[ - \frac{1}{2} \leq \sin(x - y) \leq \frac{3}{2} \]
However, since the sine function can only take values between -1 and 1, we can further restrict our analysis.

### Step 3: Use known values of sine and cosine
We know that:
\[ \sin(x - y) = \frac{1}{2} \quad \text{and} \quad \cos(x + y) = \frac{1}{2} \]
These equations imply specific angles:
- For \( \sin(x - y) = \frac{1}{2} \), the angles can be:
\[ x - y = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad x - y = \frac{5\pi}{6} + 2n\pi \]
- For \( \cos(x + y) = \frac{1}{2} \), the angles can be:
\[ x + y = \frac{\pi}{3} + 2m\pi \quad \text{or} \quad x + y = \frac{5\pi}{3} + 2m\pi \]

### Step 4: Choose appropriate values
Since we are looking for values of \( x \) and \( y \) in the range \( [0, \pi] \), we can set \( n = 0 \) and \( m = 0 \):
- From \( x - y = \frac{\pi}{6} \)
- From \( x + y = \frac{\pi}{3} \)

### Step 5: Solve the system of equations
We have the following system of equations:
1. \( x - y = \frac{\pi}{6} \) (Equation 1)
2. \( x + y = \frac{\pi}{3} \) (Equation 2)

Adding these two equations:
\[ (x - y) + (x + y) = \frac{\pi}{6} + \frac{\pi}{3} \]
This simplifies to:
\[ 2x = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]
Thus,
\[ x = \frac{\pi}{4} \]

### Step 6: Find \( y \)
Now substitute \( x \) back into one of the equations to find \( y \):
\[ \frac{\pi}{4} + y = \frac{\pi}{3} \]
Rearranging gives:
\[ y = \frac{\pi}{3} - \frac{\pi}{4} \]
Finding a common denominator (12):
\[ y = \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12} \]

### Final Values
Thus, the values of \( x \) and \( y \) are:
\[ x = \frac{\pi}{4}, \quad y = \frac{\pi}{12} \]