If x = sec θ - cos θ and y = sec n θ - cos n θ , then x 2 + 4 d y d x 2 =

Step-by-step solution

x = sec θ - cos θ ⇒ d x d θ = sec θ tan θ + sin θ y = sec n θ - cos n θ ⇒ d y d θ = n sec n θ tan θ + n cos n - 1 θ sin θ d y d x = d y d θ d x d θ = n sec n θ tan θ + cos n - 1 θ sin θ sec θ tan θ + sin θ d y d x = n 1 cos n + 1 θ + cos n - 1 θ 1 cos 2 θ + 1 x 2 + 4 d y d x 2 = 1 - cos 2 θ cos θ 2 + 4 n 2 1 + cos 2 n θ 2 1 + cos 2 θ 2 cos 2 n - 2 θ = n 2