alleryStart learning →Which product equals cos a - cos b?
A2 cos((a + b)/2) cos((a - b)/2)
B2 sin((a + b)/2) sin((a - b)/2)
C-2 sin((a + b)/2) sin((a - b)/2)
D-2 cos((a + b)/2) sin((a - b)/2)
Answer & Solution
Correct answer: C. -2 sin((a + b)/2) sin((a - b)/2)
1. A difference of cosines becomes a product of two sines carrying a leading minus sign.
2. The first sine takes the half sum of the angles and the second takes the half difference.
3. So cos a - cos b = -2 sin((a + b)/2) sin((a - b)/2).
4. Test it with a = 0 and b = pi: the left side is 1 - (-1) = 2.
5. The right side is -2 sin(pi/2) sin(-pi/2) = -2 times 1 times (-1) = 2, so the formula holds.
6. Dropping the minus sign would give -2 at the same test values, so sign is what separates the two sine options.
_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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