Prove that `f(x)=(sinx)/x` is monotonically decreasing in `[0,pi/2]dot` Hence, prove that `(2x)/pi

Explanation

Let f(X)=`(sinx)/(x)`
`therefore f(x)=(xcosx-sinx)/(x^(2))=cosx(x-tanx)/(x^(2))`
To find sign of f(x) we consider
`g(x) =x-tanx,0lt xlt (pi)/(2)`
`therefore g(x) =1 -sec^(2)lt0`
Thus , g(x) is decreasing function f.Therefore ,
`g(x)ltg(0)`
or `x-tanx lt0`
or `f(x) lt0`
Thus f(x) is a decreasing function
Also `0ltx ltpi//2`
or `f(pi//2)ltf(x)ltunderset(xrarr0)lim f(x)`
or `(2)/(pi)lt(sinx)/(x)lt1`