Answer :

The 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 = sin3θ ……equation 1


x = 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


Applying chain rule to differentiate sin3θ :


…………..equation 4


………..equation 5


Again differentiating w.r.t x:




Applying product rule and chain rule to differentiate:




[using equation 3 to put the value of dθ/dx]


Multiplying y both sides to approach towards the expression we want to prove-




[from equation 1, substituting for y]


Adding equation 5 after squaring it:





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