**Toolbox:**

- $\large \frac{dy}{dx}=\large\frac{\large\frac{dy}{d\theta}}{\large\frac{dx}{d\theta}}$
- $ 2sin\large\frac{\theta}{2}cos\large\frac{\theta}{2}=sin\theta$
- $\large \frac{d^2y}{dx^2}=\large\frac{d}{d\theta} \bigg( \large\frac{dy}{dx} \bigg) $ x $\large \frac{d\theta}{dx} $

$ \large\frac{dx}{d\theta} = -sin\theta + \large\frac{1}{tan\large\frac{\theta}{2}}$ x $ sec^2\large\frac{\theta}{2}$ x $ \large\frac{1}{2}$

$ = -sin\theta + \large\frac{1}{sin\theta}=\large\frac{cos^2\theta}{sin\theta}=cosec\theta cot\theta$

$ \large\frac{dy}{d\theta}=cos\theta$

$\large\frac{dy}{dx}=\large\frac{cos\theta}{cos^2\theta}$x$sin\theta=tan\theta$

$\large \frac{d^2y}{dx^2}=sec^2\theta$x$\large\large\frac{sin\theta}{cos^2\theta}=\large\frac{sin\theta}{cos^4\theta}$

$ \large\frac{d^2y}{dx^2}\: when \: \theta=\large\frac{\pi}{4}\: is\: \large\frac{4}{\sqrt 2}=2\sqrt2$