`"For x " in (0,(pi)/(2)) , "Prove that sin "x gt x -(x^(2))/(6)`

Explanation

`"Let " f(x) =sin x-x +(x^(3))/(6)`
`f(x) =cos x -1+(x^(2))/(2)`
we cannot decide at this point whether f(x) is positive or negative hence let us check for monotonic nature of f(x)
`f''(x) = x -sin x"`
`"Since f''(x) "gt0 " " rArr f(x)" is "M.I " " "for " x in (0,(pi)/(2))`
`rArr f(x) gt f(0) rArr f(x) gt0`
`rArr f(x) " is " M.IrArr f(x) gt f(0)`
`rArr sinx -x +(x^(3))/(6) gt 0 rArr sin x gt x-(x^(2))/(6) . " Hence Proved "`