The normal to the curve represented parametrically by $x=a(\cos \theta +\theta \sin \theta )$  and $y=a(\sin \theta -\theta \cos \theta )$  at any point $\theta$ , is such that it 

$\frac{dy}{dx}=\frac{a\{\cos \theta +\theta \sin -\cos \}}{a\{-\sin \theta +\theta \cos \theta +\sin \theta \}}$ 
 $\frac{dy}{dx}=\frac{\theta \sin \theta }{\theta \cos \theta }=\tan \theta$
Normal slope ;  $y-a\{\sin \theta -\theta \cos \theta \}$
 $=-\frac{\cos \theta }{\sin \theta }\left[x-(a\cos \theta +\theta \sin \theta \right]$
 $y\sin \theta -a{\sin }^2\theta +a\theta \sin \theta \cos \theta$
 $=-x\cos \theta +a{\cos }^2\theta +a\theta \sin \theta \cos \theta$
$x\cos \theta +y\sin \theta =a$ 
Slope of normal = $-\cot \theta$
$=\tan (\frac{\pi }{2}+\theta )$ 
normal makes an angle of  $\frac{\pi }{2}+\theta$
with x-axis