Question

Find the points on the curve where the tangent line is horizontal . x=4(cosθ-cos^2θ) , y=4(sinθ-sinθcosθ)

Answer 1

X = 4(cosθ - cos²θ) = 4cosθ - 4cos²θ
dx/dθ = -4sinθ + 8sinθcosθ = -4sinθ(1 - 2cosθ)

dy/dθ = 4(cosθ - cos²θ + sin²θ) = 4(cosθ - 1)

∴dy/dx = 4(cosθ - 1)/-4sinθ(1 - 2cosθ) = 1-cosθ/(sinθ - 2sinθcosθ)
Now, we're finding a horizontal tangent, which is when dy/dx = 0 (horizontal tangent).
1-cosθ/(sinθ - 2sinθcosθ) = 0
1-cosθ = 0
cosθ = 1

θ = 0, 360 in the interval [0, 360].
At θ = 0, x = 0, y = 0

Only points in the open interval [-2π, 2π] are at (0, 0), unless I made a mistake somewhere.