If a , b , c are sides of the triangle A B C and 1 a b 1 c a 1 b c = 0, then the value of cos ⁡ 2 A + cos ⁡ 2 B + cos ⁡ 2 C is equal to

Step-by-step solution

By expansion of the determinant, we get, 1 a b 1 c a 1 b c = 1 c 2 - a b - 1 a c - b 2 + 1 a 2 - b c = 0 ⇒ a 2 + b 2 + c 2 - a b - b c - c a = 0 which is true for a = b = c Hence, ∆ A B C must be an equilateral triangle ⇒ A = B = C = 60 o ⇒ cos ⁡ 2 A + cos ⁡ 2 B + cos ⁡ 2 C = - 3 2