The value of ${\cos }^3\mathrm{\theta }+{\cos }^3\left({120}^0+\mathrm{\theta }\right)+{\cos }^3\left(\mathrm{\theta }-{120}^0\right)$ is 

$$\begin{gathered} {\cos }^3+{\mathrm{\theta cos}}^3\hspace{0.17em}\hspace{0.17em}(\mathrm{\theta }+{120}^0)+\hspace{0.17em}\hspace{0.17em}{\cos }^3(\mathrm{\theta }-{120}^0) \\ =\hspace{0.17em}\hspace{0.17em}\frac{\cos 3\mathrm{\theta }+3\mathrm{cos\theta }}{4}+\hspace{0.17em}\hspace{0.17em}\frac{\cos (3\mathrm{\theta }+{360}^0)+3\cos (\mathrm{\theta }+{120}^0)}{4} \\ +\hspace{0.17em}\hspace{0.17em}\frac{\cos (3\mathrm{\theta }-{360}^0)+3\cos (\mathrm{\theta }-{120}^0)}{4} \\ \left(\because \hspace{0.17em}\hspace{0.17em}\cos \hspace{0.17em}\hspace{0.17em}3\mathrm{\alpha }=4{\cos }^3\mathrm{\alpha }-3\mathrm{cos\alpha }\hspace{0.17em}\hspace{0.17em}\right) \\ \therefore \hspace{0.17em}\hspace{0.17em}{\cos }^3\hspace{0.17em}\hspace{0.17em}\mathrm{\alpha }=\hspace{0.17em}\hspace{0.17em}\frac{\cos 3\mathrm{\alpha }+3\mathrm{cos\alpha }}{4} \\ =\hspace{0.17em}\hspace{0.17em}\frac{3}{4}\hspace{0.17em}\hspace{0.17em}\cos \hspace{0.17em}\hspace{0.17em}3\mathrm{\theta }+\frac{1}{4}\hspace{0.17em}\hspace{0.17em}\left\{\cos (\mathrm{\theta }+{120}^0)+\hspace{0.17em}\hspace{0.17em}\cos (\mathrm{\theta }-{120}^0)+\mathrm{cos\theta }\right\} \\ =\hspace{0.17em}\hspace{0.17em}\frac{3}{4}\hspace{0.17em}\hspace{0.17em}\cos 3\mathrm{\theta }+\frac{1}{4}\hspace{0.17em}\hspace{0.17em}\left\{2\mathrm{cos\theta cos}{120}^0+\mathrm{cos\theta }\right\} \\ =\hspace{0.17em}\hspace{0.17em}\frac{3}{4}\hspace{0.17em}\hspace{0.17em}\cos \hspace{0.17em}\hspace{0.17em}3\mathrm{\theta }+\frac{1}{4}\hspace{0.17em}\hspace{0.17em}\left\{2\mathrm{cos\theta }\left(-\frac{1}{2}\right)+\mathrm{cos\theta }\right\} \end{gathered}$$