Find maximize of the function $\frac{a}{1+a^2}+\frac{b}{1+b^2}-\frac{1}{c^2+1}$ Let $a,b\in R^+$ such that $ab+bc+ca=1$. Find the maximize of $$P=\frac{a}{1+a^2}+\frac{b}{1+b^2}-\frac{1}{c^2+1}$$

By Wolframalpha i can see that if $a=b=2-\sqrt 3;c=\sqrt 3$ we will have $P=\dfrac 1 4$
WLOG $a\le b$. I proved that $$P=f(a,b,c)\le f(a,a,c)\le 0$$
$$\Leftrightarrow -(a^2-4a+1)^2\le 0$$
My proof is based on the value of $P$ so it's inconvenient if i dont know value of $P$. This inequality is not homogeneous and symmetric so i dont any idead to solve it
Can you help me solve it without using the equality and value of $P$? 

Let $a=\tan\frac{\alpha}{2}$, $b=\tan\frac{\beta}{2}$ and $c=\tan\frac{\gamma}{2},$ where $\left\{\frac{\alpha}{2},\frac{\beta}{2},\frac{\gamma}{2}\right\}\subset(0^{\circ},90^{\circ}).$
Thus, $\alpha+\beta+\gamma=180^{\circ}$ and
$$P=\frac{1}{2}\sin\alpha+\frac{1}{2}\sin\beta-\cos^2\frac{\gamma}{2}=\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}-\cos^2\frac{\gamma}{2}=$$
$$=\cos\frac{\gamma}{2}\cos\frac{\alpha-\beta}{2}-\cos^2\frac{\gamma}{2}\leq\cos\frac{\gamma}{2}-\cos^2\frac{\gamma}{2}$$
Can you end it now?
I got that the maximal value it's $\frac{1}{4}.$
