Which trigonometric identity is equal to 2 sin A sin B?

Introduction / Context:
This question assesses your understanding of standard trigonometric product to sum identities. It is important to know how to convert products of sine and cosine functions into sums or differences for simplification and integration.

Given Data / Assumptions:

- We are asked to find an equivalent expression for 2 sin A sin B.
- A and B are angles where trigonometric functions are defined.
- We should choose the identity that matches 2 sin A sin B.

Concept / Approach:
The relevant identity from trigonometric product to sum formulas is 2 sin A sin B = cos(A − B) − cos(A + B). This identity is derived using the cosine addition and subtraction formulas. Recognizing the correct pattern is the key step in solving this question.

Step-by-Step Solution:
Step 1: Recall the product to sum identity: cos C − cos D = −2 sin((C + D)/2) sin((C − D)/2).Step 2: A more directly useful identity is 2 sin A sin B = cos(A − B) − cos(A + B).Step 3: Compare the expression 2 sin A sin B with the given options. We are looking for a difference of cosines with arguments A − B and A + B.Step 4: Option b is cos(A − B) − cos(A + B), which matches the required identity exactly.Step 5: Therefore, 2 sin A sin B equals cos(A − B) − cos(A + B).

Verification / Alternative check:
To verify, pick simple values for A and B, for example A = 60 degrees and B = 30 degrees. Then 2 sin A sin B = 2 sin 60° sin 30° = 2 * (√3/2) * (1/2) = √3/2. Now compute cos(A − B) − cos(A + B) = cos 30° − cos 90° = (√3/2) − 0 = √3/2. Both expressions give the same result, confirming the identity.

Why Other Options Are Wrong:
- sin(A + B) + sin(A − B) corresponds to the identity 2 sin A cos B, not 2 sin A sin B.
- sin(A + B) − sin(A − B) corresponds to 2 cos A sin B, which is not the same product of sines.
- cos(A + B) + cos(A − B) corresponds to 2 cos A cos B, another different identity.
- cos(A + B) − cos(A − B) does not match the standard pattern for 2 sin A sin B and gives the negative of the correct expression.

Common Pitfalls:
Students often mix up the product to sum identities, confusing whether a result should be a sum or a difference or swapping sine and cosine roles. A good way to remember is that products of two sines turn into differences of cosines, while products of sine and cosine or two cosines follow different patterns.

Final Answer:
The correct identity is 2 sin A sin B = cos(A − B) − cos(A + B).