alleryStart learning →Simplify (cos(4t) - cos(2t)) / (sin(4t) + sin(2t)) with the sum-to-product formulas.
Atan(t)
B-cot(t)
Ccot(t)
D-tan(t)
Answer & Solution
Correct answer: D. -tan(t)
1. The numerator is a difference of cosines, so it becomes -2 sin((4t + 2t)/2) sin((4t - 2t)/2) = -2 sin(3t) sin(t).
2. The denominator is a sum of sines, so it becomes 2 sin((4t + 2t)/2) cos((4t - 2t)/2) = 2 sin(3t) cos(t).
3. The factor 2 sin(3t) is common to both and cancels.
4. What remains is -sin(t) / cos(t).
5. By the quotient identity that equals -tan(t).
6. Choosing tan(t) drops the minus sign that the cosine difference formula carries.
7. Choosing a cotangent inverts the surviving ratio, which would need sine and cosine to swap places.
_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_
Keep practising Trigonometric Identities and Equations
Browse all Trigonometric Identities and Equations questions for B.Sc Mathematics, or explore more Mathematics topics.
Related questions
If a ladder's base is a feet from the wall and its length is 4a feet, how high up the wallSafety rules place a ladder's base 1 foot from the wall for every 4 feet of ladder length.Compared with y = sin t, the graph of y = sin(2t) is which transformation?Solving sin(n t) = c across one turn requires going around the unit circle how many times?Solve cos(2t) = 1/2 on the interval from 0 to 2 pi.Solve 3 cos t + 3 = 2 sin^2 t exactly on the interval from 0 to 2 pi.Solve cos t cos(2t) + sin t sin(2t) = sqrt(3)/2 on the interval from 0 to 2 pi.Solve 2 sin^2 t + sin t = 0 exactly on the interval from 0 to 2 pi.