If $x=a\left(2\cos \theta +\cos 2\theta \right),y=a\left(2\sin \theta +\sin 2\theta \right)$ , then $\frac{dy}{dx}$  is

$\frac{dx}{d\theta }=a\left[-2\sin \theta -2\sin 2\theta \right]$

$=-2a\left[\sin \theta +\sin 2\theta \right]$

Given $y=a\left(2\sin \theta +\sin 2\theta \right)$

$\Rightarrow \frac{dy}{d\theta }=a\left[2\cos \theta +2\cos 2\theta \right]$

$=2a\left[\cos \theta +\cos 2\theta \right]$

$\frac{dy}{dx}=\frac{2a\left(\cos \theta +\cos 2\theta \right)}{-2a\left(\sin \theta +\sin 2\theta \right)}$

$=-\frac{2\cos \frac{3\theta }{2}\cos \frac{\theta }{2}}{2\sin \frac{3\theta }{2}\cos \frac{\theta }{2}}$

$=-\cot \left(\frac{3\theta }{2}\right)$