$\begin{align}  & x=a(\theta -\sin \theta ) \\ & \frac{dx}{d\theta }=a(1-\cos \theta ),\text{   }\frac{d\theta }{dx}=\frac{1}{a(1-\cos \theta )} \\ & y=a(1-\cos \theta ),\text{ }\frac{dy}{d\theta }=a\sin \theta  \\ & \frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{d\theta }{dx}=a\sin \theta \times \frac{1}{a(1-\cos \theta )} \\ & \frac{dy}{dx}=\frac{\sin \theta }{1-\cos \theta } \\ & 1+{{\left( \frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{\sin \theta }{1-\cos \theta } \right)}^{2}}=1+\frac{{{\sin }^{2}}\theta }{1-2\cos \theta +{{\cos }^{2}}\theta } \\ & 1+{{\left( \frac{dy}{dx} \right)}^{2}}=1+\frac{1-{{\cos }^{2}}\theta }{{{(1-\cos \theta )}^{2}}}=1+\frac{(1-\cos \theta )(1+\cos \theta )}{{{(1-\cos \theta )}^{2}}} \\ & 1+{{\left( \frac{dy}{dx} \right)}^{2}}=1+\frac{1+\cos \theta }{1-\cos \theta }=\frac{1-\cos \theta +1+\cos \theta }{1-\cos \theta } \\ & 1+{{\left( \frac{dy}{dx} \right)}^{2}}=\frac{2}{1-\cos \theta }=\frac{2}{1-(1-2{{\sin }^{2}}\tfrac{\theta }{2})}=\frac{2}{2{{\sin }^{2}}\frac{\theta }{2}} \\ & 1+{{\left( \frac{dy}{dx} \right)}^{2}}={{\operatorname{cosec}}^{2}}\frac{\theta }{2} \\\end{align}$