If `x=a(cos 2 theta+2 theta sin 2 theta) " and" y=a(sin 2 theta - 2 theta cos 2 theta)`, find `(d^(2)y)/(dx^(2)) " at" theta =(pi)/(8)`.

Explanation

To find \(\frac{d^2y}{dx^2}\) at \(\theta = \frac{\pi}{8}\) given the equations \(x = a(\cos 2\theta + 2\theta \sin 2\theta)\) and \(y = a(\sin 2\theta - 2\theta \cos 2\theta)\), we will follow these steps:

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

1. Differentiate \(x\):
\[ \frac{dx}{d\theta} = a\left(-2\sin 2\theta + 2\sin 2\theta + 4\theta \cos 2\theta\right) \]
...