Mathematics

This problem hinges on the constraint $A + B + C = 180°$, which means $C = 180° - (A + B)$. The key insight is to use the complementary angle relationship: $\cos C = -\cos(A + B)$ and $\sin C = \sin(A + B)$. Once you rewrite the angles using this constraint, the three fractions will combine through a common denominator $\sin A \sin B \sin C$, and the numerator will simplify dramatically using the cosine addition formula and trigonometric identities.

This problem hinges on the constraint $A + B + C = 180°$, which means $C = 180° - (A + B)$. The key insight is to use the complementary angle relationship: $\cos C = -\cos(A + B)$ and $\sin C = \sin(A + B)$. Once you rewrite the angles using this constraint, the three fractions will combine through a common denominator $\sin A \sin B \sin C$, and the numerator will simplify dramatically using the cosine addition formula and trigonometric identities.