If x = cos θ, y = sin³θ. Prove that

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: