<s>[INST] Trigonometric inequality $\tan\left(\frac{1}{1+x^2}\right)<\frac{1}{1-x+x^2}$ for all $x>1/2$ In Hobson's book "A treatise on plane trigonometry", the following exercise is to be found

If $x>1/2$ then 
  $$\frac{1}{1+x+x^2}<\tan\left(\frac{1}{1+x^2}\right)<\frac{1}{1-x+x^2}.$$

The first inequality is easy as it follows directly from $\alpha<\tan(\alpha)$ and the fact that $x$ is positive.
For the second one can produce the even tighter bound (on the given interval, checked only graphically)
$$\tan(\alpha)<\frac{\alpha}{1-\frac{\alpha^2}{2}}$$
but from here I cannot get to the exact given inequality.
I am aware of the fact that $\tan(\alpha)\sim \alpha$ for small $\alpha$, but I cannot manage to find the exact bound given in the exercise.
Any hints are welcome.
 [/INST] Taking arctangent of both sides (and using its monotonicity in a given range):
$$f_1(x)=\frac{1}{1+x^2} \\ f_2(x)=\arctan \frac{1}{1-x+x^2}$$
$$f_2(x)= \int_0^1 \frac{1-x+x^2}{(1-x+x^2)^2+t^2} dt$$
We need to prove that:
$$\int_0^1 \frac{(1-x+x^2)(1+x^2)}{(1-x+x^2)^2+t^2} dt >^? 1$$
$$\int_0^1 \frac{1-x+2x^2-x^3+x^4}{(1-2x+3x^2-2x^3+x^4)+t^2} dt >^? 1$$
$$1+\int_0^1 \frac{x(1-x+x^2)-t^2}{(1-x+x^2)^2+t^2} dt >^? 1$$
Now we need to prove that:
$$f_3(x)=\int_0^1 \frac{x(1-x+x^2)-t^2}{(1-x+x^2)^2+t^2} dt >^? 0, \quad x> \frac12$$
Since $1-x+x^2 \geq 1$ for $x \geq 1$, the inequality follows trivially in this case, so we only need to consider the range:
$$\frac12 <x<1$$
$$f_3'(x)=\int_0^1 \frac{(1-x^2)(1-x+x^2)^2-(1-4 x+3 x^2-4 x^3) t^2}{((1-x+x^2)^2+t^2)^2} dt$$
We have:
$$(1-x^2)(1-x+x^2)^2-(1-4 x+3 x^2-4 x^3)=x (2-x) (1+x^2)^2 >0, \quad 0<x<2$$
So $f_3(x)$ increases monotonely in the range of interest, which means it's enough to prove that:
$$f_3 \left( \frac{1}{2} \right) >^? 0$$
$$f_3 \left( \frac{1}{2} \right)=\int_0^1 \frac{\frac{3}{8}-t^2}{\frac{9}{16}+t^2} dt=\frac{3}{4} \int_0^{4/3} \frac{2/3-u^2}{1+u^2} du= \\ = \frac{5}{4} \arctan \frac{4}{3}-1 >0$$

This finishes the proof.

Note:
$$\arctan \frac{4}{3}= 2 \arctan \frac{3}{4} \left(\sqrt{1+\frac{16}{9}}-1 \right)=2 \arctan \frac{1}{2}$$
$$f_3 \left( \frac{1}{2} \right)=\frac{5}{2} \arctan \frac{1}{2}-1>\frac{5}{2} \left(\frac12-\frac{1}{24} \right)-1=\frac{7}{48}$$
Thus we don't need any numerical checks.
</s>