Question

If x and y are connected parametrically by the equation, without eliminating the parameter, find dy/dx x=cosθ - cos2θ , y=sinθ - sin2θ

Answer 1

Given, $\bf{x=cos\theta-cos2\theta,y=sin\theta-sin2\theta}$

$\bf{\underline{x=cos\theta-cos2\theta}}$
now, differentiate x with respect to θ
dx/dθ = d(cosθ - cos2θ)/dθ
= -sinθ + 2sin2θ ----(1)

$\bf{\underline{x=sin\theta-sin2\theta}}$
now differentiate y with respect to θ
dy/dθ = d(sinθ - sin2θ)/dθ
= cosθ - 2cos2θ ------(2)

dividing equations (2) by (1),
{dy/dθ}/{dx/dθ} = (2sin2θ - sinθ)/(cosθ - 2cos2θ)
dy/dx = (2sin2θ - sinθ)/(cosθ - 2cos2θ)