$Periodofthefunctionf\left(x\right)=\cos \left(2\pi \left\{2x\right\}\right)+\sin \left(2\pi \left\{2x\right\}\right)is\left(where\left\{x\right\}denotesthefractionalpartofx\right).$

$$\begin{gathered} f\left(x\right)=\cos \left(2\pi \left\{2x\right\}\right)+\sin \left(2\pi \left\{2x\right\}\right) \\ = \cos \left(2\pi \left(2x-\left[2x\right]\right)\right)+\sin \left(2\pi \left(2x-\left[2x\right]\right)\right) \\ =\cos \left(4\pi x-2\pi \left[2x\right]\right)+\sin \left(4\pi x-2\pi \left[2x\right]\right) \\ =\cos \left(4\pi x\right)-\sin \left(4\pi x\right) \\ period of \cos \left(4\pi x\right)=\frac{2\pi }{4\pi }=\frac{1}{2} \\ period of \sin \left(4\pi x\right)=\frac{2\pi }{4\pi }=\frac{1}{2} \\ L.C.M is \frac{1}{2} \end{gathered}$$