What is the value of `{ sin (90 + x) cos (pi - (x-y))] } + {cos (90 + x)sin [ pi - (x - y)]}`?

Options

A. `-cosy`
B. `-siny`
C. cos x
D. tan y

Answer

To solve the expression \( \{ \sin(90^\circ + x) \cos(\pi - (x - y)) \} + \{ \cos(90^\circ + x) \sin(\pi - (x - y)) \} \), we can use the sine addition formula.

### Step-by-step Solution:

1. **Identify the angles**:
- Let \( A = 90^\circ + x \)
- Let \( B = \pi - (x - y) \)

2. **Use the sine addition formula**:
The sine addition formula states that:
\[ \sin A \cos B + \cos A \sin B = \sin(A + B) \]
Therefore, we can rewrite our expression as:
\[ \sin(90^\circ + x) \cos(\pi - (x - y)) + \cos(90^\circ + x) \sin(\pi - (x - y)) = \sin((90^\circ + x) + (\pi - (x - y))) \]

3. **Simplify the angle**:
Now, we simplify the angle:
\[ (90^\circ + x) + (\pi - (x - y)) = 90^\circ + x + \pi - x + y = 90^\circ + \pi + y \]
Since \( 90^\circ + \pi = 180^\circ \):
\[ 180^\circ + y \]

4. **Calculate the sine**:
Now we find:
\[ \sin(180^\circ + y) \]
Using the sine property, we know:
\[ \sin(180^\circ + y) = -\sin(y) \]

5. **Final Result**:
Therefore, the value of the original expression is:
\[ -\sin(y) \]

### Final Answer:
\[ -\sin(y) \]