Explanation

(a)
Given,
$x = a(\theta + \sin\theta)$ and $y = a(1 - \cos\theta)$
$\Rightarrow \ \ \ \frac{dx}{d\theta} = a(1 + \cos\theta)$ and $\frac{dy}{d\theta} = a\sin\theta$
$\therefore\ \ \ \ \frac{dy}{dx} = \frac{a\sin\theta}{a(1 + \cos\theta)} = \tan\frac{\theta}{2}$
Now, length of subtangent$= \left| \frac{y}{dy/dx} \right|$
$\therefore\ \ \ \ ST = \frac{a(1 - \cos\theta)}{\tan{(\theta/2)}} = a\sin\theta$
$\Rightarrow$ Length of suntangent at $\theta = \frac{\pi}{2}$,
$ST = a\sin\frac{\pi}{2} = a$
And length of subnormal$= \left| y\frac{dy}{dx} \right|$
$\Rightarrow \ \ \ \ \ SN = a\left( 1 - \cos\theta \right).\tan\frac{\theta}{2}$
$= a.2\sin^{2}\frac{\theta}{2}\tan\frac{\theta}{2}$
$\Rightarrow$ Length of subnormal at $\theta = \frac{\pi}{2}$,
$SN + a.2.1.2 = a$
Hence, $SN = ST$