If `sin(60^(@)-x)=cos(y+60^(@))` then the value of `sin(x-y)` is

Options

A. `(1)/(sqrt(2))`
B. `(1)/(2)`
C. `(sqrt(3))/(2)`
D. 1

Answer

To solve the equation \( \sin(60^\circ - x) = \cos(y + 60^\circ) \) and find the value of \( \sin(x - y) \), we can follow these steps:

### Step 1: Use the co-function identity
We know that \( \sin(\theta) = \cos(90^\circ - \theta) \). Thus, we can rewrite the left side of the equation:
\[ \sin(60^\circ - x) = \cos(90^\circ - (60^\circ - x)) = \cos(30^\circ + x) \]
So, we can rewrite the equation as:
\[ \cos(30^\circ + x) = \cos(y + 60^\circ) \]

### Step 2: Set the angles equal
Since the cosines are equal, we can set the angles equal to each other:
\[ 30^\circ + x = y + 60^\circ \quad \text{or} \quad 30^\circ + x = - (y + 60^\circ) + 360^\circ \]
For simplicity, we will consider the first case:
\[ 30^\circ + x = y + 60^\circ \]

### Step 3: Rearrange to find \( x - y \)
Rearranging the equation gives us:
\[ x - y = 60^\circ - 30^\circ \]
Thus,
\[ x - y = 30^\circ \]

### Step 4: Find \( \sin(x - y) \)
Now that we have \( x - y = 30^\circ \), we can find \( \sin(x - y) \):
\[ \sin(x - y) = \sin(30^\circ) \]
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]

### Final Answer
Therefore, the value of \( \sin(x - y) \) is:
\[ \boxed{\frac{1}{2}} \]