Maximize $P=\frac{1}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+y^{2}}}+\frac{1}{\sqrt{1+z^{2}}}$ For $x,y,z$ are positive real numbers that satisfy $xy+yz+xz=1$. Maximize $$P=\frac{1}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+y^{2}}}+\frac{1}{\sqrt{1+z^{2}}}.$$

I think if we let $x=\tan A;y=\tan B;z=\tan C$, then  
$$P\Leftrightarrow \displaystyle \text {sin}A+\text {sin}B+ \text{sin}C\leq \frac{3}{2}. \hspace{2cm}(1)$$
But I can't prove $(1)$. 

Hint : You're making a mistake. 
You have to prove this 
$$
\cos A + \cos B + \cos C \le \frac{3}{2}
$$
because :
$$
1+\tan^2 \theta = \sec^2 \theta
$$
