In X-Y plane, the path defined by the equation `(1)/(x^(m))+(1)/(y^(m)) +(k)/((x+y)^(n)) =0`, is a hyperbola if `m =1, k =- 1, n=0`

Explanation

`f(x) =(4sin x-2x-x cos x )/(2+cosx)`
`=(4sinx)/(2+cosx)-(2x+xcosx)/(2+cosx)=(4sinx)/(2+cosx)-x`
`therefore f(x) =(4cosx(2+cosx)-(-sinx)(4 sin x))/(2+cos)^(2)`
f(x)=0
cosx=0
`x= (pi)/(2),(3pi)/(2)`
`cos x l gt 0 for x in (0,pi//2)` in `(3pi//2,2pi)`
and `cos x lt 0` for x in `(pi//2,pi)cup(pi,3pi//2)`
`=(4+8cosx)/(2+cosx)^(2)-1`
`=(4+8cosx-(2+cosx)^(2))/(2+cosx)^(2)`