If x = 2 sin θ - sin 2 θ and y = 2 cos θ - cos 2 θ , θ ∈ 0 , 2 π , then d 2 y d x 2 at θ = π is:

Step-by-step solution

d x d θ = 2 cos θ - 2 cos 2 θ d y d θ = - 2 sin θ + 2 sin 2 θ ∴   d y d x = sin 2 θ - sin θ cos θ - cos 2 θ = 2 sin θ 2 . cos 3 θ 2 2 sin θ 2 . sin 3 θ 2 = cot 3 θ 2 d 2 y d x 2 = d d θ d y d x d θ d x = - 3 2 cosec 2 3 θ 2 . d θ d x ⇒ d 2 y d x 2 = - 3 2 cosec 2 3 θ 2 2 cos θ - cos 2 θ ⇒ d 2 y d x 2 θ = π = 3 4 - 1 - 1 = 3 8