If x = a (1 – cos θ), y =a (θ + sin 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,

y = a (θ + sin θ) ……equation 1

x = 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.

…..equation 3

Similarly,

……equation 4

[∵

…..equation 5

Differentiating again w.r.t x :

Using product rule and chain rule of differentiation together:

[using equation 4]

As we have to find

∴ put θ = π/2 in above equation:

=