<s>[INST] Prove $\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$ If $a.b,c \in  \mathbb{R^+}$ and $ab+bc+ca=1$ 
Then Prove $$S=\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$$
My try we have $$S=\sum \frac{a}{\sqrt{a^2+ab+bc+ca}}=\sum \frac{a}{\sqrt{a+b}\sqrt{a+c})}$$
any hint here?
 [/INST] Not in the spirit you want to g, but here is alternative solution:
Let $x = \arctan {1\over a}$, $y = \arctan {1\over  b}$ and $z = \arctan {1\over c}$ 
Then $a= \cot x$ and so $\frac{a}{\sqrt{a^2+1}} = \cos x\sin x$ ....
Now we have to prove $$\sin 2x+\sin 2y+\sin 2z\leq 3$$ which is true.
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