If a, b, c are the sides of triangle ABC, then what is | a² b sinA c sinA ; b sinA 1 cosA ; c sinA cosA 1 | equal to?

Using the sine rule, a/sinA = b/sinB = c/sinC = 2R, so b sinA = a sinB and c sinA = a sinC. Multiplying the first row by sinA and using these relations leads to two proportional rows after factoring. More directly, expanding the determinant and using a² = b² + c² – 2bc cosA along with sinA relations causes all terms to cancel, giving zero.