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Which of the following trigonometric identities is correct?

1. (sin 2A + sin 2B)/(sin 2A - sin 2B) = tan(A + B)/tan(A - B)
2. (cos 2B + cos 2A)/(cos 2B - cos 2A) = cot(A + B) cot(A - B)
3. cos(A + B) + sin(A - B) = 2 sin(45 deg + A) cos(45 deg + B)
4. (sin 7θ - sin 5θ)/(cos 7θ + cos 5θ) = tan θ

Correct answer: (sin 2A + sin 2B)/(sin 2A - sin 2B) = tan(A + B)/tan(A - B)

Solution

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

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