The period of $\sin \left(\frac{\pi }{3}-x\right)\sin \left(\frac{\pi }{3}+x\right)$  is

$=\frac{\cos \left(\frac{\pi }{3}-x-\frac{\pi }{3}-x\right)-\cos \left(\frac{\pi }{3}-x+\frac{\pi }{3}+x\right)}{2}$

$=\frac{\cos \left(-2x\right)-\cos \left(\frac{\pi }{2}+30°\right)}{2}$

$=\frac{\cos \left(-2x\right)-\sin 30°}{2}$

$\therefore$ Period of $\cos \left(-2x\right)=\frac{2\pi }{\left|-2\right|}=\pi$