If the line x = y = z intersects the line x sin A + y sin B + z sin C - 18 = 0 = x sin 2 A + y sin 2 B + z sin 2 C - 9 , where A , B , C are the angles of a triangle A B C , then 80 sin A 2 sin B 2 sin C 2 is equal to _________.

Step-by-step solution

Given, The line x = y = z intersects the line x   sin A + y   sin B + z   sin C - 18 = 0 = x   sin 2 A + y   sin 2 B + z sin 2 C - 9 So, let x = y = z = k Now putting the value in x   sin A + y   sin B + z   sin C = 18 we get, k sin A + sin B + sin C = 18 Now we know that, if A ,   B   &   C are angles of triangle then sin A + sin B + sin C = 4 cos A 2 · cos B 2 · cos C 2 ⇒ k 4 cos A 2 · cos B 2 · cos C 2 = 18   . . . . . i Al