Using the standard sum to product identity, if the trigonometric expression sin C + sin D is rewritten as a product, what is the correct equivalent product form in terms of the angles C and D?

Introduction / Context:
This question checks understanding of trigonometric sum to product identities. These identities are useful in simplifying expressions involving sums of sine or cosine functions and often appear in aptitude exams and competitive mathematics questions. Here, we need to convert a sum of two sine terms into an equivalent product form involving sines and cosines of half sum and half difference of the angles.

Given Data / Assumptions:

• We are given the expression sin C + sin D.
• C and D are angles measured in degrees or radians, but the identity is independent of the unit.
• We assume that all trigonometric functions involved are defined for the chosen values of C and D.

Concept / Approach:
The standard sum to product identity for sine states that sin C + sin D can be rewritten as a product using half sum and half difference of the angles. The exact identity is sin C + sin D = 2 sin((C + D)/2) cos((C - D)/2). The idea is that replacing a sum by a product can make some integrals, equations, or simplifications easier to handle. We simply match this standard form to the options given.

Step-by-Step Solution:
Recall the identity: sin C + sin D = 2 sin((C + D)/2) * cos((C - D)/2).Compare this with the options. We look for a product of a sine of half the sum and a cosine of half the difference.Option b is 2sin[(C + D)/2]cos[(C - D)/2], which exactly matches the identity.None of the other options have both the correct functions and the correct argument structure for half sum and half difference.Therefore, the correct product form that represents sin C + sin D is option b.

Verification / Alternative check:
To verify, choose simple angles, for example C = 30 degrees and D = 90 degrees. Then sin C + sin D = sin 30 + sin 90 = 0.5 + 1 = 1.5. Now evaluate the right hand side: 2 sin((C + D)/2) cos((C - D)/2) = 2 sin(60) cos(-30). sin 60 is sqrt(3)/2 and cos(-30) equals cos 30 which is sqrt(3)/2. The product becomes 2 * (sqrt(3)/2) * (sqrt(3)/2) = 2 * 3/4 = 3/2 = 1.5. The values match, confirming the identity.

Why Other Options Are Wrong:

• Option a places cosine on the half sum and sine on the half difference, which corresponds to a different identity, sin C - sin D.
• Option c has cos on both factors and does not represent the sum of two sine functions.
• Option d uses sine for both factors, which again does not match the standard sum to product identity for sines.
• Option e leaves the expression as sin(C + D), which is unrelated to the required product identity.

Common Pitfalls:

• Confusing the identities for sin C + sin D and sin C - sin D, which swap sine and cosine roles.
• Mixing up the half sum and half difference in the arguments, since their positions matter.
• Assuming that a sum of sines converts to a single sine of a sum without the product structure.

Final Answer:
2sin[(C + D)/2]cos[(C - D)/2]