If A² + B² + C² = 0, then what is the value of the following? |1 cosC cosB; cosC 1 cosA; cosB cosA 1|

A² + B² + C² = 0 with real numbers implies A = B = C = 0 only in real case, but here interpreted as a constraint. If A = B = C = 0, all cosines equal 1, so the determinant is |1 1 1; 1 1 1; 1 1 1| = 0. Hence value is 0.