Let [x] = greatest integer $\le$x. Define $\mathrm{f}:\mathbf{R}\to \mathbf{R}$ by  $\mathrm{f}\left(\mathrm{x}\right)=\cos \left(5\mathrm{x}\right)\cos \left[5\mathrm{x}\right]+\sin \left(5\mathrm{x}\right)\sin \left[5\mathrm{x}\right]$ then period of f is

Rewrite f as

$\mathrm{f}\left(\mathrm{x}\right)=\cos (5\mathrm{x}-[5\mathrm{x}\left]\right)$

Note that

$\begin{array}{rl}\mathrm{f}(\mathrm{x}+\frac{1}{5})& =\cos (5(\mathrm{x}+\frac{1}{5})-\left[5(\mathrm{x}+\frac{1}{5})\right])\\ & =\cos (5\mathrm{x}+1-[5\mathrm{x}+1\left]\right)\\ & =\cos (5\mathrm{x}+1-(\left[5\mathrm{x}\right]+1\left)\right)\\ & =\cos (5\mathrm{x}-[5\mathrm{x}\left]\right)=\mathrm{f}\left(\mathrm{x}\right)\end{array}$

Thus, f has a period of 1/5