alleryStart learning →How is the product-to-sum formula for sin a sin b derived?
ABy adding the two cosine formulas together
BBy adding the two sine formulas together
CBy subtracting the two sine formulas
DBy subtracting the two cosine formulas
Answer & Solution
Correct answer: D. By subtracting the two cosine formulas
1. The two cosine formulas differ only in the sign in front of sin a sin b.
2. Subtracting cos(a + b) from cos(a - b) cancels the cos a cos b terms.
3. What survives is 2 sin a sin b on one side and cos(a - b) - cos(a + b) on the other.
4. Dividing by 2 gives sin a sin b = (1/2)[cos(a - b) - cos(a + b)].
5. Adding the same two cosine formulas would isolate cos a cos b instead, so the sign of the operation is what separates the two results.
_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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