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If A + B + C = 180 deg, prove that sin²(A/2) + sin²(B/2) + sin²(C/2) = 1 - 2*sin(A/2)*sin(B/2)*sin(C/2). Which expression equals sin²(A/2) + sin²(B/2) + sin²(C/2)?

1. 1 - 2*sin(A/2)*sin(B/2)*sin(C/2)
2. 1 + 2*sin(A/2)*sin(B/2)*sin(C/2)
3. 1 - 2*cos(A/2)*cos(B/2)*cos(C/2)
4. 2*sin(A/2)*sin(B/2)*sin(C/2)

Correct answer: 1 - 2*sin(A/2)*sin(B/2)*sin(C/2)

Solution

Converting to cosines and using A + B + C = 180 deg reduces the sum to 1 - 2*sin(A/2)*sin(B/2)*sin(C/2).

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