Using the standard product-to-sum trigonometric identities, simplify the expression 2 cos A * sin B. If 2 cos A * sin B = x, which equivalent expression correctly represents x?

Introduction / Context:
This trigonometry question focuses on the product-to-sum identities, which convert products of sine and cosine functions into sums or differences of sines or cosines. These identities are very useful in simplifying expressions, solving trigonometric equations, and working with integrals in calculus. Here, you are asked to simplify 2 cos A * sin B and express it in terms of sine functions using the correct identity.

Given Data / Assumptions:

• The expression is 2 cos A * sin B.
• A and B are real angles (no special restriction on quadrants is needed for the identity itself).
• We define x = 2 cos A * sin B.
• We must express x using a sum or difference of sine functions based on a standard identity.

Concept / Approach:
The standard product-to-sum identities include: 2 sin X * cos Y = sin(X + Y) + sin(X − Y) and symmetric variants. In this problem, the factors are arranged as cos A * sin B, but multiplication is commutative, so we can think of the product as 2 sin B * cos A. Then we can apply the identity with X = B and Y = A. After that, we may need to rewrite terms like sin(B − A) in a form involving A − B, but we must be careful with sign changes because sine is an odd function: sin(−θ) = −sin θ.

Step-by-Step Solution:
1) Start with x = 2 cos A * sin B. 2) Rewrite the product using commutativity: x = 2 sin B * cos A. 3) Apply the identity 2 sin X * cos Y = sin(X + Y) + sin(X − Y) with X = B and Y = A. 4) This gives x = sin(B + A) + sin(B − A). 5) Since addition is commutative, B + A = A + B, so sin(B + A) = sin(A + B). 6) Thus x = sin(A + B) + sin(B − A). 7) Note that sin(B − A) = −sin(A − B), because sine is an odd function. 8) Substitute sin(B − A) = −sin(A − B) to obtain x = sin(A + B) − sin(A − B).

Verification / Alternative check:
You can test the identity with specific angle values, for example A = 30 degrees and B = 60 degrees. Compute the left side: 2 cos 30 * sin 60 = 2 * (√3 / 2) * (√3 / 2) = 2 * (3/4) = 3/2. Now compute the right side for option a: sin(A + B) − sin(A − B) = sin 90 − sin(−30). Using standard values, sin 90 = 1 and sin(−30) = −1/2, so the expression becomes 1 − (−1/2) = 1 + 1/2 = 3/2. The two sides match, confirming that the identity is correct for this choice of A and B, and supporting option a as the correct expression.

Why Other Options Are Wrong:
Option b, sin(A + B) + sin(A − B), corresponds to 2 sin A * cos B instead of 2 cos A * sin B. Option c and option d involve cosine functions and match the product-to-sum identities for cos X * cos Y, not sin and cos products. Option e rearranges terms in a way that introduces a sign pattern inconsistent with the underlying identity. Only option a correctly represents 2 cos A * sin B as a difference of sine terms.

Common Pitfalls:
A common error is to mis remember which product corresponds to which sum identity, for example confusing the formulas for 2 sin X * cos Y and 2 cos X * cos Y. Another pitfall is failing to notice that sin(B − A) and sin(A − B) differ by a sign, leading to an incorrect choice between a sum and a difference. Writing down the general identity first and then substituting X and Y carefully helps avoid these mistakes.

Final Answer:
Using the product-to-sum identity, the expression 2 cos A * sin B is equal to sin(A + B) - sin(A - B).