If x = a (θ + sin θ), y = a (1+ cos θ), prove that .

Idea of parametric form of differentiation:

If y = f (θ) and x = g(θ) i.e. y is a function of θ and x is also some other function of θ.

Then dy/dθ = f’(θ) and dx/dθ = g’(θ)

We can write :

Given,

x = a (θ + sin θ) ……equation 1

y = a (1+ cos θ) ……equation 2

to prove :

We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find

As,

So, lets first find dy/dx using parametric form and differentiate it again.

[∵ from equation 2] …..equation 3

Similarly,

……equation 4

[∵

[∵ from equation 2] …..equation 5

Differentiating again w.r.t x :

Using product rule and chain rule of differentiation together:

[using equation 3 and 5]

[ from equation 1]

∴ ….proved