Using the relation `2(1-cosx)

Explanation

Give that `2(1-cosx)ltx^(2),xne0`
To prove sin `(tanx)gex,x in [0,pi//4]`, let us consider
f(x)=sin(tanx )-x
f(x) =cos (tanx ) `sec^(2)x-1`
`=(cos(tanx)-cos^(2)x)/(cost(2))x`
`sin^(2)x(1-(1))/(2cos^(2)x(cost(2))x`
`=sin^(2)x(cos2x)/(2cos^(4))x gt 0 forall x in [0,[pi//4]`
Therefore `f(x)gt0` .Thus f(x) is an increasing function
so ,for x in `[0,pi//4]`we have
`xge0` or `f(x) gef(0)`
`sin(tanx)-xgesin(tan 0)-0`
`sin(tan x)-xge0`
`sin (tan x)gex`