Three Variables-Inequality with $a+b+c=abc$ $a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$
Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$

Hint：Let $a=\tan\alpha, b=\tan\beta, c=\tan\gamma$, then $a+b+c=abc$ implies $\alpha+\beta+\gamma=\pi$. So the objective function becomes $$\cos\alpha+\cos\beta+\cos\gamma=\cos\alpha+\cos\beta-\cos(\alpha+\beta)$$
