If T is the period of the function $\mathrm{f}\left(\mathrm{x}\right)=\left[8\mathrm{x}+7\right]+\left|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}\right|-8\mathrm{x}$  (where [.] denotes the greatest integer function), then the value of T  is ______

$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right)=\left[8\mathrm{x}+7\right]+\left|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}\right|-8\mathrm{x}\\ =\left[8\mathrm{x}\right]-8\mathrm{x}+7+\left|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}\right|\\ =-\left\{8\mathrm{x}\right\}+\left|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}\right|+7\\ \mathrm{Period} \mathrm{of} \left\{8\mathrm{x}\right\} \mathrm{is} 1/8.\\ \mathrm{Also}, \left|\tan 2\mathrm{\pi x}+\cot 2\mathrm{\pi x}\right|=\left|\frac{\sin 2\mathrm{\pi x}}{\cos 2\mathrm{\pi x}}+\frac{\cos 2\mathrm{\pi x}}{\sin 2\mathrm{\pi x}}\right|\\ =\left|\frac{1}{\sin 2\mathrm{\pi xcos}2\mathrm{\pi x}}\right|=\left|2 \mathrm{cosec} 4\mathrm{\pi x}\right|\end{array}$

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

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