The period of $\sin \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{12}+\cos \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{4}+\tan \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{3},$Where $\left[x\right]$ represents the greatest integer less than or equal x is  

Since,
 $$\begin{gathered} \sin \frac{\mathrm{\pi }\left[\mathrm{x}+24\right]}{12}=\sin \frac{\mathrm{\pi }}{12}\left(24+\left[\mathrm{x}\right]\right) \\ =\sin \left(2\mathrm{\pi }+\frac{\mathrm{\pi }\left[\mathrm{x}\right]}{2}\right)=\sin \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{12} \end{gathered}$$
The period of $\sin \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{12} \mathrm{is} 24.$
Similarly, period of $\cos \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{4}$ is 8 and period of $\tan \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{3}=3$.
Hence, the period of the given function $=\mathrm{LCM} \mathrm{of} 24,8,3=24.$