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Which of the following trigonometric identities is correct?

1. (cos2B + cos2A)/(cos2B - cos2A) = cot(A + B) cot(A - B)
2. (sin2A + sin2B)/(sin2A - sin2B) = tan(A + B)/tan(A - B)
3. (cos3A - cosA)/(sin3A - sinA) + (cos2A - cos4A)/(sin4A - sin2A) = sinA/(cos2A cos3A)
4. cos(A + B) + sin(A - B) = 2 sin(45 deg + A) cos(45 deg + B)

Correct answer: (cos2B + cos2A)/(cos2B - cos2A) = cot(A + B) cot(A - B)

Solution

Using sum-to-product, the ratio becomes [2cos(A+B)cos(A-B)]/[2sin(A+B)sin(A-B)] = cot(A+B)cot(A-B), so (a) is a valid identity.

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