If `x=acos^(3)t,y=asin^(3)t,"show that "(dy)/(dx)=-((y)/(x))^((1)/(3))`

Explanation

`x=acos^(3)t,y=asin^(3)t`
Differentiating w.r.t. t, we get,
`(dx)/(dt)=a(d)/(dt)(cost)^(3)=a.3(cost)^(2)(d)/(dt)(cost)`
`=3acos^(2)t(-sint)=-3acos^(2)tsint`
`and(dy)/(dt)=a(d)/(dt)(sint)^(3)=a.3(sint)^(2)(d)/(dt)(sint)`
`=3asin^(2)t.cost`
`therefore(dy)/(dx)=(dy)/(dt)//(dx)/(dt)=(3asin^(2)tcost)/(-3acos^(2)tsint)=-(sint)/(cost)" ... (1)"`
Now `x=acos^(3)t thereforecos^(3)t=(x)/(a)thereforecost=((x)/(a))^(1//3)`
`y=asin^(3)t thereforesin^(3)t=(y)/(a) thereforesint=((y)/(a))^(1//3)`
`therefore "from (1)",(dy)/(dx)=-(y^(1//3)//a^(1//3))/(x^(1//3)//a^(1//3))=-((y)/(x))^(1//3)`