Discuss monotonocity of `f(x)=x/(sinx)a n d` `g(x)=x/(tanx),w h e r e

Answer

f(x) is increasing and g(x) decreasing

Explanation

`f(x) =(sinx -x cos x )/(sin^(2)x) =cos x(tanx-x)/(sin^(2)x)`
`0ltx le1 or x` in first quadreant for which tan `x gt x, cos x gt 0` or `f'(x) gt for 0 lt x le 1`
Thus f(x) is an increasing function
`(g(x) =(tan x -x sec^(2))x)/(tan^(2)x)=(sinx cos x-x)/(sin^(2)x)=(sin 2x-2x)/(2 sin^(2) x)`
Now `0lt2xle2 for which sin 2xlt2x`
or `g'(x) lt0`
Thus g(x) is decreasing.