Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the expression for cosαcosβ in terms of sums and differences?

1/2(cos(α+β) + cos(α-β))
1/2(cos(α-β) - cos(α+β))
sinαsinβ
(cosα + cosβ)/2

The correct answer is: 1/2(cos(α+β) + cos(α-β))

The expression for cosαcosβ can be transformed into sums and differences by using a trigonometric identity. Specifically, the product-to-sum formulas are utilized to express the product of cosines in terms of sums. The correct answer employs the product-to-sum identity for the cosine function, which states that: cosαcosβ = 1/2 [cos(α + β) + cos(α - β)]. This identity shows how the product of two cosine functions can be represented as the average of the cosine of their sum and the cosine of their difference. This transformation is particularly useful for simplifying problems in trigonometry, allowing for the management of expressions involving products of cosines by converting them into sums. The other choices do not correctly apply this identity or involve other relationships that are not equivalent to the product of cosα and cosβ.