Definition of period: f(x+T)=f(x) Substitute the function expression into the formula, take the special value to see if there is a reasonable period, and then take the minimum reasonable value of t..

xcosx =(x+T)cos(x+T)； Let x=0, 0cos0=0=TcosT (from this formula: T=0 (truncation) or cosT=0, t = npai); cos(x+npai)= cos xcos(npai)-sinx sin(npai)=-sinx sin(npai)； Taking n as odd number and even number respectively cannot meet the definition, so the function has no periodicity.

[What I want to ask here is (sin (x)) 2] sin2x = sin2 (x+t); Let x=0, sin0 = 0 = sin2 (t), t = npaisin (x+npai) = sinxcos (npai)+cosxsin (npai) = sinxcos (npai); Cos (npai) 2 = 1, so sin (x+npai) 2 = sin (x) 2. To sum up, T=npai, and the smallest is pai.

Actually, SIN 2x can be calculated by sum-difference integration, Y = SIN 2x = (1-COS2x)/2, and the period of COS2x is pai, so the period of SIN 2x is also pai.