The period of the function $\mathrm{f}\left(\mathrm{x}\right)=[8\mathrm{x}+7]+\mid \tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}\mid -8\mathrm{x}$ (where [.] denotes the greatest integer function), is _______.

$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right)=[8\mathrm{x}+7]+|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}|-8\mathrm{x}\\ =\left[8\mathrm{x}\right]-8\mathrm{x}-7+|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}|\\ =-\left\{8\mathrm{x}\right\}+|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}|+7\end{array}$

Period is $\left\{8\mathrm{x}\right\}\text{ is }1/8\text{. }$

Also, $|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}|=\left|\frac{\sin 2\mathrm{\pi x}}{\cos 2\mathrm{\pi x}}+\frac{\cos 2\mathrm{\pi x}}{\sin 2\mathrm{\pi x}}\right|$

$=\left|\frac{1}{\sin 2\mathrm{\pi xcos}2\mathrm{\pi x}}\right|=|2\mathrm{cosec}4\mathrm{\pi x}|$

Now, period of $2 \mathrm{cosec} 4\mathrm{\pi x}\text{ is }1/2$. Then period of $|2\mathrm{cosec}4\mathrm{\pi x}|\text{ is }1/4$.

Therefore, period is L.C.M. of $\frac{1}{8}\text{ and }\frac{1}{4}\text{ which is }\frac{1}{4}\text{. }$