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\(\begin{align}
  & \cos \left( \alpha +\beta  \right)=\cos \alpha +\cos \beta  \\
& \cos \alpha \cos \beta -\sin \alpha \sin \beta -\cos \alpha -\cos \beta =0 \\
&  \\
& \cos \beta =x,\sin \beta =y \\
& {{x}^{2}}+{{y}^{2}}=1 \\
& \left( \cos \alpha -1 \right)x-(\sin \alpha )y-\cos \alpha =0 \\
&  \\
& \frac{\left| -\cos \alpha  \right|}{\sqrt{{{\left( \cos \alpha -1 \right)}^{2}}+{{\left[ -(\sin \alpha ) \right]}^{2}}}}\le 1 \\
& \frac{\cos \alpha }{\sqrt{2-2\cos \alpha }}\le 1 \\
& -1\le \cos \alpha \le -1+\sqrt{3} \\
\end{align}\)

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