If `x=costheta` , `y=sin^3theta` , prove that `y\ (d^2y)/(dx^2)+((dy)/(dx))^2=3\ sin^2theta(5\ cos^2theta-1)` .

Explanation

To prove the equation \( y \frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta (5 \cos^2 \theta - 1) \) given \( x = \cos \theta \) and \( y = \sin^3 \theta \), we will follow these steps:

### Step 1: Find \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \)
Given \( y = \sin^3 \theta \):
\[ \frac{dy}{d\theta} = 3 \sin^2 \theta \cdot \cos \theta \]
Given \( x = \cos \theta \):
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