$ = \frac{{n{{\sec }^n}\theta \tan \theta + n{{\cos }^{n - 1}}\theta \sin \theta }}{{\sec \theta \tan \theta + \sin \theta }}$
$ = \frac{{n({{\sec }^n}\theta + {{\cos }^n}\theta )}}{{\sec \theta + \cos \theta }}$ (Dividing ${N^r}$ and ${D^r}$ by $\tan \theta $)
==> ${\left( {\frac{{dy}}{{dx}}} \right)^2} = \frac{{{n^2}{{({{\sec }^n}\theta + {{\cos }^n}\theta )}^2}}}{{{{(\sec \theta + \cos \theta )}^2}}}$
$ = \frac{{{n^2}[{{({{\sec }^n}\theta - {{\cos }^n}\theta )}^2} + 4{{\sec }^n}\theta {{\cos }^n}\theta ]}}{{{{(\sec \theta - \cos \theta )}^2} + 4\sec \theta .\cos \theta }} = \frac{{{n^2}({y^2} + 4)}}{{{x^2} + 4}}$
==> $({x^2} + 4){\rm{ }}{\left( {\frac{{dy}}{{dx}}} \right)^2} = {n^2}({y^2} + 4)$.
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