If the trigonometric sum cos C + cos D is rewritten using the standard sum to product identity, what is the equivalent product expression in terms of angles C and D?

Introduction / Context:This problem focuses on trigonometric sum to product identities. Converting sums like cos C + cos D into product form is common in trigonometry, signal processing, and exam questions that test formula recall and correct substitution skills.

Given Data / Assumptions:

• We are given the sum cos C + cos D.
• Angles C and D are within a domain where cosine is defined.
• We need to express this sum in a standard product form using half sum and half difference of the angles.

Concept / Approach:The standard sum to product identity for cosines is cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2). This identity is distinct from the one for sines and from the difference identity for cosine. The key is to remember which trigonometric function appears in each factor. For cos C + cos D, both factors are cosines, one with half the sum and the other with half the difference of the angles.

Step-by-Step Solution:Recall the identity: cos C + cos D = 2 cos((C + D)/2) * cos((C - D)/2).Examine each option to see which one matches this structure.Option a is 2cos[(C + D)/2]cos[(C - D)/2], which is exactly the identity we need.Options b and d involve sine functions, which correspond to sum or difference identities for sine rather than cosine.Therefore, the correct product form for cos C + cos D is option a.

Verification / Alternative check:To verify, choose convenient angles, for example C = 60 degrees and D = 0 degrees. Then cos C + cos D = cos 60 + cos 0 = 0.5 + 1 = 1.5. Now evaluate the right hand side of the identity: 2 cos((C + D)/2) cos((C - D)/2) = 2 cos(30) cos(30). cos 30 is sqrt(3)/2, so the product becomes 2 * (sqrt(3)/2) * (sqrt(3)/2) = 2 * 3/4 = 3/2 = 1.5. The values match, confirming that the identity is correct.

Why Other Options Are Wrong:

• Option b uses sine functions for both factors, which corresponds to the identity for cos C - cos D and not for cos C + cos D.
• Option c mixes cosine and sine and does not match any standard sum to product identity for cos C + cos D.
• Option d also mixes sine and cosine but in the opposite pattern and is not appropriate here.
• Option e simply writes cos(C + D) and does not transform the sum into a product at all.

Common Pitfalls:

• Confusing the identity for cos C + cos D with cos C - cos D or sin C ± sin D.
• Swapping half sum and half difference in the wrong positions in the arguments.
• Assuming that a sum of cosines directly becomes a single cosine without the factor of 2 and the product structure.

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