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If A + B + C = 180 deg, prove that sin²(A/2) + sin²(B/2) - sin²(C/2) = 1 - 2 cos(A/2) cos(B/2) sin(C/2).

1. Identity is true (proved)
2. Identity is false
3. RHS should be 1 - 2 sin(A/2) sin(B/2) cos(C/2)
4. RHS should be 1 + 2 cos(A/2) cos(B/2) sin(C/2)

Correct answer: Identity is true (proved)

Solution

Converting to cosines gives the LHS = 1/2 - (1/2)(cos A + cos B - cos C). Using sum-to-product and A+B+C = 180 deg reduces it to 1 - 2 cos(A/2)cos(B/2)sin(C/2).

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