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If x = cos θ, y = sin3θ. 

Prove that \(y\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2\)\(=3sin^2\theta(5cos^2\theta-1)\)

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 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 :\(\frac{dy}{dx}\)\(=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\)

Given,

y = sin3θ ……equation 1

x = cos θ ……equation 2

To prove:   \(y\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2\)\(=3sin^2\theta(5cos^2\theta-1)\) 

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 \(\frac{d^2y}{dx^2}\)

As, \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)

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

Applying chain rule to differentiate sin3θ :

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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