$x = a(\theta - \sin \theta ),y = a(1 + \cos \theta )$

Here,$x = a(\theta - \sin \theta )$ and …(i)
$y = a(1 + \cos \theta )$ …(ii)
Differentiating (i) \& (ii) w.r.t. $\theta$, we get

$\cfrac{{dx}}{{d\theta }} = a[1 - \cos \theta ]$ and

$\cfrac{{dy}}{{d\theta }} = a[ - \sin \theta ] = - a\sin \theta$

therefore, $\cfrac{{dy}}{{dx}} = \cfrac{{dy/d\theta }}{{dx/d\theta }} = \cfrac{{ - a\sin \theta }}{{a(1 - \cos \theta )}} = \cfrac{{ - \sin \theta }}{{1 - \cos \theta }}$

$= \cfrac{{ - 2\sin \theta /2\cos \theta /2}}{{2{{\sin }^2}\theta /2}} = - \cot \cfrac{\theta }{2}$