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If A + B + C = 180 deg, prove that sin(A/2) + sin(B/2) + sin(C/2) - 1 = 4 sin((pi - A)/4) sin((pi - B)/4) sin((pi - C)/4). Which statement is correct?

1. It is a valid identity
2. RHS should use cos instead of sin
3. LHS should be sin(A/2)+sin(B/2)+sin(C/2)+1
4. It is false for an equilateral triangle

Correct answer: It is a valid identity

Solution

Using A/2+B/2+C/2 = pi/2 and product-to-sum on sin(A/2)+sin(B/2), then combining with sin(C/2)-1, the expression factors into the stated product of sines of (pi-A)/4 etc.

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