Let A = x 4 + 4 x 3 + 2 x 2 - 4 x + 7 where x = cot ⁡ 11 π 8 and B = 1 - cos ⁡ 8 θ tan 2 ⁡ 4 θ + 1 + cos ⁡ 8 θ cot 2 ⁡ 4 θ where θ = 9 o , then the value of A × B 2 is equal to

Step-by-step solution

A = x 4 + 4 x 3 + 2 x 2 - 4 x + 7 where x = cot ⁡ 11 π 8 = cot ⁡ π + 3 π 8 = cot ⁡ 3 π 8 = cot ⁡ π 2 - π 8 = tan ⁡ π 8 = 2 - 1 i.e. x = 2 - 1 ⇒ x 2 = 2 + 1 - 2 2 = 1 - 2 2 - 1 ⇒ x 2 = 1 - 2 x ⇒ x 2 + 2 x - 1 = 0 So, A = x 2 + 2 x - 1 2 + 6 = 6 B = 2 sin 2 ⁡ 4 θ tan 2 ⁡ 4 θ + 2 cos 2 ⁡ 4 θ cot 2 ⁡ 4 θ = 2 cos 2 ⁡ 4 θ + sin 2 ⁡ 4 θ = 2 ⇒ A B = 12 ⇒ A B 2 = 6