If `x=cos e ctheta-sinthetaa n dy=cos e c^ntheta-sin^ntheta,` then show that `(x^2+4)((dy)/(dx))^2=n^2(y^2+4)dot`

Explanation

Given, ` x = sec theta-cos theta and y = sec ^(n) theta-cos^(n) theta`
On differentiating w.r.t. ` theta` respectively, we get
`(dx)/(d theta) = sec theta tan theta + sin theta`
and ` (dy)/(d theta) = n sec^(n-1) theta* sec theta tan theta - n cos^(n-1) theta* (-sin theta)`
`rArr (dx)/(d theta) = tan theta (sec theta + cos theta)`
`and (dy)/(d theta) = n tan theta(sec^(n) theta + cos^(n) theta)`
`rArr (dy)/(dx) = (n (sec^(n)theta+cos^(n) theta))/(sec theta + cos theta)`
`:. ((dy)/(dx))^(2) =(n^(2)(sec^(n)theta+cos^(n) theta)^(2))/((sec theta+ cos theta)^(2))`
`=(n^(2){(sec^(n) theta-cos^(n) theta)^(2)+4})/({(sec theta - cos theta)^(2)+4})=(n^(2)(y^(2)+4))/((x^(2)+4))`
` rArr (x^(2) + 4)((dy)/(dx))^(2) = n^(2) (y^(2)+4)`