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Which of the following trigonometric identities is correct? (Choose the valid identity.)

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

Correct answer: (sin 7T - sin 5T) / (cos 7T + cos 5T) = tan T

Solution

For the first option, numerator = 2 cos 6T sin T and denominator = 2 cos 6T cos T, so the ratio is tan T, which is correct.

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