Prove that `(tan^(-1)(1/e))^(2)+(2e)/(sqrt(e^(2)+1)` `lt (tan^(-1) e)^2 + (2)/(sqrt(e^(2)+1)`

Explanation

we have to prove
`tan^(-1)(1)/(C )sqrt(e^(2)+1)lt(tan^(-1)e)^(2)+(32)/(e^(2)+1)`
or `tan^(-1)(1)/(e )^(2)+(2)/sqrt(1)(e )^(2)+1lt(tan^(-1)e)^(2)+(2)/sqrt(e^(2)+1)`
Now f(x) =`(2tan^(-1)x)/(1+x^(2))-(2x)/(x^(2)+1)`
It is difficult to study the sign of f(x)
so we let g(x) =`tn^(-1)x-(x)/sqrt(x^(2)+1)`
g(x) is and increasing function
for `xgt 0,g(x)gtg(0)`
`g(x)gt0`
`f(x)gt0`
f(x) is increasing function
Since `(1)/(e )lte`, we have `f(1/e)ltf(x)`