The expression has the form #2sin(A)cos(A)# where #A=2x#.
#sin(2A)=2sin(A)cos(A)#, so we can rewrite the given expression as #sin(4x)#. The period of a sine function is #P=(2pi)/B#, where #B# is the coefficient of #x#, so the period is #(2pi)/4=pi/2#.