alleryStart learning →How are the sum-to-product formulas obtained?
AFrom the product-to-sum identities, using substitution
BFrom the half-angle identities, using a square root
CFrom the reduction formulas, using a power change
DFrom the cofunction identities, using complements
Answer & Solution
Correct answer: A. From the product-to-sum identities, using substitution
1. Take the product-to-sum formula sin a cos b = (1/2)[sin(a + b) + sin(a - b)].
2. Substitute a = (u + v) / 2 and b = (u - v) / 2, so that a + b = u and a - b = v.
3. The formula becomes sin((u + v)/2) cos((u - v)/2) = (1/2)[sin u + sin v].
4. Multiplying by 2 gives sin u + sin v as a product, which is a sum-to-product formula.
5. Half-angle and reduction formulas change powers and angles but never turn a sum of two different angles into a product.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.4 Sum-to-Product and Product-to-Sum Formulas_
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