if:`f(x)=(sinx)/(sqrt(1+tan^2x))-(cosx)/(sqrt(1+cot^2x)),` then find the range of `f(x)`

Explanation

`f(x)=(sin x)/(|sec x|)-(cos x)/(|"cosec"x|)=sin x |cos x|- cos x|sin x|`
Clearly, the domain of f(x) is `R~{(n pi)/(2),n in Z}` and the period of `f(x)` is `2 pi`.
`f(x)={(0", "x in (0,pi//2)),(-sin2x", "x in (pi//2,pi)),(0", "x in (pi,3pi//2)),(sin2x", "x in (3pi//2,2pi)):}`
The range of `f(x)` is `[-1,1].`