Explanation

(a)
$x = a(1 + \cos{\theta),\ \ \ \ \ y = a(\theta + \sin{\theta)}}$
$\frac{dx}{d\theta} = - {a\ sin}{\theta,}\ \frac{dy}{d\theta} = a\ (1 + \cos{\theta)}$
$\therefore\ \ \frac{dy}{dx}\ = \ \frac{1 + \cos\theta}{- \sin\theta}$ = $\frac{2\cos^{2}{\ \frac{\theta}{2}}}{- 2\sin{\ \frac{\theta}{2}\cos{\ \frac{\theta}{2}}}}$
$\Rightarrow \ \frac{dy}{dx} = \ - \cot\frac{\theta}{2}$
$\therefore\ \ \frac{d^{2}y}{dx^{2}} = \frac{d}{dx}\left( \frac{dy}{dx} \right) = \frac{d}{d\theta}\left( \frac{dy}{dx} \right).\frac{d\theta}{dx}$
$= \frac{d}{d\theta}\ \left( - \cot\frac{\theta}{2} \right).\frac{1}{- a\sin\theta}$
$= \frac{1}{2}cosec^{2}\frac{\theta}{2}.\frac{1}{- a\sin\theta}$
$\ \ \therefore\ \ {\ \ \ \left( \frac{d^{2}y}{dx^{2}} \right)}_{\theta = \frac{\pi}{2}} = \frac{1}{2}.2.\frac{1}{- a} = - \frac{1}{a}\$