How to show that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is strictly increasing for $x \geq 1$? I am trying to prove that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is a strictly increasing function for $x \geq 1$. 
I try to do this by showing that $f'(x)>0$ for all $x \geq 1$. 
We have $$f'(x)= \arctan(x \sin^{2} \frac{1}{x}) + \frac{x}{1+x^{2} \sin^{4} \frac{1}{x}} \left(\frac{-2\cos\frac{1}{x} \sin \frac {1}{x}}{x} + \sin^{2} \frac{1}{x}\right),$$
so it's not clear if this quantity is positive.
Now I can show that $x \arctan \left(\frac{1}{x}\right)$ is strictly increasing (by computing its derivative and using simple trig inequalities to show that it is positive), and that $$x \sin^{2} \left(\frac{1}{x}\right)- \frac{1}{x}>0$$ So my aim is somehow to compare $f(x)$ with $x\arctan \frac{1}{x}$ and use the above inequality to prove that $f'(x)>0$.
But now I am stuck. Any suggestions as to how to proceed? Or any other approach to the problem would be appreciated!

Simple, use derivative, not approximations. You get: 
$(x\arctan(x\sin^2(\frac{1}{x})))'=\arctan(x\sin^2(\frac{1}{x}) + \frac{x}{1+x^2\sin^4(\frac{1}{x})}(sin^2(\frac{1}{x}) + \frac{x}{1+x^2\sin^4(\frac{1}{x})}(-2x\sin\frac{1}{x}\cos\frac{1}{x}(-\frac{1}{x^2}))$ 
so all are positive for $x \geq 1$.
