What is the value of this trigonometric function?

Question

Consider $f:\mathbb R\to \mathbb R$ defined by $$f(x)= \arccos \Bigl(\frac{1-x^2}{1+x^2}\Bigr)+ \arcsin \Bigl(\frac{2x}{1+x^2}\Bigr)$$

What is $f(-\sqrt{3}) + f(-\ln(2)) + f(1) + f(\ln(3))$ ?

I calculated $f(-\sqrt{3})=\frac{\pi}3$ and $f(1)=\pi$, but I don't know how to calculate the other terms.

Answer 1

Using Principal values,

let $\arctan x= y,-\dfrac\pi2<y<\dfrac\pi2$

$$f(x)=\arccos(\cos2y)+\arcsin(\sin2y)=\begin{cases}4y &\mbox{if }0\le2y\le\dfrac\pi2 \\2y+\pi-2y & \mbox{if }2y>\dfrac\pi2\\-2y+(-\pi-2y) &\mbox{if } 2y<-\dfrac\pi2\end{cases}$$