If `x=a(costheta+thetasintheta)` , `y=a(sintheta-thetacostheta)` , prove that `(d^2x)/(d theta^2)=a(costheta-thetasintheta)`,\ `(d^2y)/(d theta^(2))=a(sintheta+thetacostheta)` and `(d^2y)/(dx^2)=(s e c^3theta)/(a\ theta)` .

Explanation

To solve the problem, we need to differentiate the given expressions for \( x \) and \( y \) with respect to \( \theta \) and then find the second derivatives. Let's go through the steps systematically.

### Step 1: Differentiate \( x \) with respect to \( \theta \)

Given:
\[ x = a(\cos \theta + \theta \sin \theta) \]
...