Answer :

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:


=



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