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

Here, $x = a(\cos \theta + \theta \sin \theta )$ …..(1)
and $y = a(\sin \theta - \theta \sec \theta )$ .…(2)
Differentiating (1) \& (2) w.r.t. $\theta$, we get

$\cfrac{{dx}}{{d\theta }} = a[ - \sin \theta + \theta \cdot \cos \theta + \sin \theta ] = a\theta \cos \theta$
$\cfrac{{dy}}{{d\theta }} = a[\cos \theta - (\theta ( - \sin \theta ) + \cos \theta )]$

$= a[\cos \theta + \theta \sin \theta - \cos \theta ] = a\theta \sin \theta$

therefore, $\cfrac{{dy}}{{dx}} = \cfrac{{dy/d\theta }}{{dx/d\theta }} = \cfrac{{a\theta \sin \theta }}{{a\theta \cos \theta }} = \tan \theta$