To simplify the expression [tex]\(\sin(x-y) - \sin(x+y)\)[/tex], we use trigonometric identities. We will apply the sum-to-product identities for sine functions:
1. Sum-to-Product Identities:
[tex]\[
\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)
\][/tex]
However, here we are dealing with [tex]\(\sin(x-y) - \sin(x+y)\)[/tex].
2. Applying the Specific Values:
Let [tex]\( A = x - y \)[/tex] and [tex]\( B = x + y \)[/tex]
According to the identity:
[tex]\[
\sin(x - y) - \sin(x + y) = 2 \cos \left( \frac{(x - y) + (x + y)}{2} \right) \sin \left( \frac{(x - y) - (x + y)}{2} \right)
\][/tex]
Simplify the terms inside the cos and sin functions:
[tex]\[
\cos \left( \frac{(x - y) + (x + y)}{2} \right) = \cos \left( \frac{2x}{2} \right) = \cos(x)
\][/tex]
[tex]\[
\sin \left( \frac{(x - y) - (x + y)}{2} \right) = \sin \left( \frac{-2y}{2} \right) = \sin(-y) = -\sin(y)
\][/tex]
3. Combining the Results:
[tex]\[
\sin(x - y) - \sin(x + y) = 2 \cos(x) \cdot -\sin(y)
\][/tex]
Therefore:
[tex]\[
\sin(x - y) - \sin(x + y) = -2 \sin(y) \cos(x)
\][/tex]
Thus, the correct simplification of [tex]\(\sin(x - y) - \sin(x + y)\)[/tex] is [tex]\(-2 \sin(x) \cos(y)\)[/tex].
4. Conclusion:
The corresponding multiple-choice answer is:
[tex]\[ \boxed{-2 (\sin x)(\cos y)} \][/tex]
