alleryStart learning →Evaluate cos 15 degrees minus cos 75 degrees using the sum-to-product formula.
Asqrt(2) / 2
Bsqrt(3) / 2
C-sqrt(2) / 2
Dsqrt(2) / 4
Answer & Solution
Correct answer: A. sqrt(2) / 2
1. Use cos a - cos b = -2 sin((a + b)/2) sin((a - b)/2) with a = 15 degrees and b = 75 degrees.
2. The half sum is (15 + 75)/2 = 45 degrees.
3. The half difference is (15 - 75)/2 = -30 degrees.
4. So the expression is -2 sin(45 degrees) sin(-30 degrees).
5. Substituting sin 45 = sqrt(2)/2 and sin(-30) = -1/2 gives -2 times (sqrt(2)/2) times (-1/2).
6. The two minus signs cancel and the result is sqrt(2)/2.
7. Forgetting that sine of a negative angle is negative would leave the answer at -sqrt(2)/2, which contradicts cos 15 being larger than cos 75.
_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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