The range of the function $\cos \left(\pi \left[x\right]\right)$ is

$$\begin{gathered} \left[x\right]\text{ = n ∈Z ,so cos }\left[ x\right]\text{ π =cos nπ =1 , if n is even } \\ = -1 , if n is odd \\ R\text{ange}=\{-1,1\} \end{gathered}$$