If `x=a\ s e c^3theta` and `y=a\ t a n^3theta` , find `(dy)/(dx)` at `theta=pi/3`

Explanation

`"We have "x=a sec^(3) theta and y= atan^(3)theta`
`therefore" "(dx)/(d""theta)=3a sec^(2)theta(d)/(d""theta)(sec theta)=3a sec^(3)theta tan theta`
`"and "(dy)/(d""theta)=3a tan^(2)theta(d)/(d""theta)(tan theta)=3a tan^(2)theta sec^(2) theta`
`"or "(dy)/(dx)=(dy//d""theta)/(dx//d""theta)=(3a tan^(2) theta sec^(2) theta)/(3a sec^(3)theta tan theta)=(tan theta)/(sec theta)=sin theta`
`"or "(dy)/(dx)_(theta=pi//3)=sin""(pi)/(3)=(sqrt(3))/(2)`