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Verify the identity (cos(3A) - cos(A))/(sin(3A) - sin(A)) + (cos(2A) - cos(4A))/(sin(4A) - sin(2A)) = sin(A)/(cos(2A)*cos(3A)). Which trigonometric tool makes both fractions collapse to tangent/cotangent terms?

1. Sum-to-product (factor) formulae for sums and differences of sines and cosines
2. Product-to-sum formulae only
3. The double angle formula for tan only
4. The half-angle substitution t = tan(A/2)

Correct answer: Sum-to-product (factor) formulae for sums and differences of sines and cosines

Solution

Applying sum-to-product to each numerator and denominator turns the first fraction into -cot(2A) and the second into a tan term, and combining gives sin(A)/(cos(2A)*cos(3A)).

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