Suppose \ (xcos \, x \) in cycles, according to a periodic function of the law, available
\ [\ begin {aligned} xcos \, x & = (x + T) cos \, (x + T) \\ & = (x + T) cos \, xcos \, Tsin \, xsin \, T \\ & = xcos \, xcos \, T - xsin \, xsin \, T + Tcos \, xcos \, T - Tsin \ , xsin \, T \\ \ end
{aligned} \] formula need to set up, the \ (cos \, T = 1 and Tcos \, xcos \, T- Tsin \, xsin \, T-xsin \, xsin \ , T = 0 \) ,
It found that only the formula \ (T = 0 \) , the two conditions are established only, so \ (xcos \, x \) function does not cycle