If sin C − sin D = x, express x in terms of trigonometric functions of (C + D)/2 and (C − D)/2.

Introduction / Context:
This question tests your knowledge of trigonometric sum to product identities. These identities allow you to rewrite sums and differences of sine and cosine functions in terms of products involving half angles, which is very useful in simplification and integration problems.

Given Data / Assumptions:

- Expression: sin C − sin D
- This expression is equal to x
- C and D are angles where the trigonometric functions are defined
- We must write x in terms of (C + D)/2 and (C − D)/2

Concept / Approach:
The standard identity for the difference of sines is sin C − sin D = 2 cos((C + D)/2) sin((C − D)/2). The idea is to treat C and D symmetrically, average them, and half their difference, then use this identity to transform the difference into a product. This representation is very helpful in many advanced trigonometric manipulations.

Step-by-Step Solution:
Step 1: Recall the sum to product identity: sin C − sin D = 2 cos((C + D)/2) sin((C − D)/2).Step 2: Recognize that this identity is derived by applying the sine addition and subtraction formulas in reverse and grouping terms.Step 3: By direct application, we substitute the angles C and D into the identity. Thus, sin C − sin D equals 2 cos((C + D)/2) sin((C − D)/2).Step 4: Therefore, x, which equals sin C − sin D, is given by x = 2 cos((C + D)/2) sin((C − D)/2).

Verification / Alternative check:
You can verify this identity by choosing simple values, for example C = 60 degrees and D = 30 degrees. Then sin C − sin D = sin 60° − sin 30° = (√3/2) − (1/2). The right hand side becomes 2 cos((60° + 30°)/2) sin((60° − 30°)/2) = 2 cos 45° sin 15°. Using approximate values, cos 45° ≈ 0.707 and sin 15° ≈ 0.259, so the product 2 * 0.707 * 0.259 ≈ 0.366, which is close to √3/2 − 1/2 ≈ 0.366, confirming the identity.

Why Other Options Are Wrong:
- 2 sin((C + D)/2) cos((C − D)/2) is the identity for sin C + sin D, not for sin C − sin D.
- 2 cos((C + D)/2) cos((C − D)/2) corresponds to a cosine sum, not a sine difference.
- 2 sin((C + D)/2) sin((D − C)/2) differs by a sign change and does not match the correct expression.
- 2 sin C cos D is a different identity and does not involve the half sum and half difference of C and D.

Common Pitfalls:
Students often confuse the identities for sum and difference of sines and cosines, mixing up where the sine or cosine belongs. Another pitfall is forgetting that the sign in the second factor depends on whether we have a sum or a difference. Memorizing the patterns and practicing with actual angle values helps solidify these identities.

Final Answer:
The value of x is 2 cos((C + D)/2) sin((C − D)/2).