alleryStart learning →Write 2 cos(7x/2) cos(3x/2) as a sum.
Acos(2x) - cos(5x)
Bcos(5x) - cos(2x)
Csin(2x) + sin(5x)
Dcos(2x) + cos(5x)
Answer & Solution
Correct answer: D. cos(2x) + cos(5x)
1. Use cos a cos b = (1/2)[cos(a - b) + cos(a + b)] with a = 7x/2 and b = 3x/2.
2. The leading 2 cancels the factor of 1/2, so the answer is cos(a - b) + cos(a + b).
3. The difference is 7x/2 - 3x/2 = 4x/2 = 2x.
4. The sum is 7x/2 + 3x/2 = 10x/2 = 5x.
5. So the product becomes cos(2x) + cos(5x).
6. A product of cosines turns into a sum of cosines, so the option built from sines cannot be right.
_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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