$\text{ The period of }{\sin }^{6}x+{\cos }^{6}x\text{ is }$

$\begin{array}{l}{\sin }^{6}x+{\cos }^{6}x\\ =\left({\sin }^{2}x+{\cos }^{2}x\right)\left({\sin }^{4}x+{\cos }^{4}x-{\sin }^{2}x{\cos }^{2}x\right)\\ =1-3{\sin }^{2}x{\cos }^{2}x=1-\frac{3}{4}{\sin }^{2}2x=1-\frac{3}{8}(1-cos4x)\\ {\sin }^{6}x+{\cos }^{6}x=\frac{5+3\cos 4x}{8}\\ period of{\sin }^{6}x+{\cos }^{6}x=period of \cos 4x\\ period=\frac{2\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{2}\\ \end{array}$