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 [x]represents the greatest integer less than or equal to x is

Since,

$\begin{array}{rl}\sin \frac{\mathrm{\pi }[\mathrm{x}+24]}{12}& =\sin \frac{\mathrm{\pi }}{12}(24+[\mathrm{x}\left]\right)\\ & =\sin (2\mathrm{\pi }+\frac{\mathrm{\pi }\left[\mathrm{x}\right]}{2})=\sin \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{12}\end{array}$

The period of $\sin \frac{\mathrm{\pi }\left[\mathrm{x}\right]}{12}$ 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 = LCM of 24,8,3=24