Write $a,b,c$ instead of $\alpha,\beta,\gamma$. Let $p(t)$ be the polynomial in question. Let 
$$t_*:=\frac{1}{30} \left(15-\sqrt{15} \sqrt{4 \left(\cos ^2(a)+\cos
   ^2(b)+\cos ^2(c)\right)+3}\right).$$
Then (under the given conditions on $a,b,c$) 
$$p(t_*)\ge0. \tag{1}\label{1}$$
Also, 
$$p(1/5)\ge0\text{ or }t_*\le1/5. \tag{2}\label{2}$$
So, either $p(1/5)\ge0$ or $t_*\le1/5\ \&\ p(t_*)\ge0$, so that $\max_{t\le1/5} p(t)\ge0$. Also, $p(-\infty+)=-\infty<0$. 

Thus, $p(t)=0$ for some real $t\le1/5$. $\quad\Box$

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Claims \eqref{1} and \eqref{2} can be verified algorithmically, since $t_*$, $p(t_*)$, and the conditions on $a,b,c$ are algebraic in $\cos a,\cos b,\cos c$. 

This has been done with Mathematica, which took about 0.25 sec to check \eqref{1} and about 0.15 sec to check \eqref{2}. Here is an image of the Mathematica notebook: 

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/eA4U0QDv.png