If A, B and C are the angles of a triangle, then sin2A + sin2B + sin2C equals:

Pair the first two: sin2A + sin2B = 2·sin(A + B)·cos(A − B) = 2·sinC·cos(A − B), because sin(A + B) = sin(180° − C) = sinC. Also sin2C = 2·sinC·cosC = −2·sinC·cos(A + B). Factoring out 2sinC leaves 2sinC[cos(A − B) − cos(A + B)] = 2sinC × 2·sinA·sinB. Check with A = 90°, B = 60°, C = 30°: the sum is 0 + 0.866 + 0.866 = 1.732 and 4(1)(0.866)(0.5) = 1.732. The coefficient is 4, not 2.