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If x=a\ s e c^3theta and y=a\ t a n^3the...

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

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`x= asec^2 theta`
`y = a tan^3 theta`
`dx/(d theta)= 3a^2 sec^2 theta d/(d theta) sec theta`
`= 3a^2 sec^2 theta sec theta tan theta`
`= 3asec^2 theta tan theta`
now,`dy/(dtheta) = 3atan^2 theta d/(d theta) (tan theta) = 3atan^2theta sec^2 theta`
so, `(dy)/(dx) = (dy)/(d theta) xx (d theta)/(dx) `
`= (3atan^2 theta sec^2theta)/(3asec^2 theta tan theta) `
`= (tan theta)/(sec theta) = sin theta`
at `theta = pi/3`
`= sin pi/3 = sqrt3/2`
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