Among the following functions defined on R into R, the constant function is

$$\begin{gathered} f\left(x\right)={\cos }^{2}x+{\cos }^2\left(x+\frac{\pi }{3}\right)+\sin x\cdot \sin \left(x+\frac{\pi }{3}\right) \\ =1+{\cos }^2\left(x+\frac{\pi }{3}\right)- {\sin }^{2}x+\frac{1}{2}2\sin x\cdot \sin \left(x+\frac{\pi }{3}\right) \\ =1+\cos \left(2x+\frac{\mathrm{\pi }}{3}\right)\cos \frac{\mathrm{\pi }}{3}+\frac{1}{2}\left(\cos (2x+\frac{\mathrm{\pi }}{3})-\cos \frac{\pi }{3} \right) \\ =1+\frac{1}{2}\cos \left(2x+\frac{\mathrm{\pi }}{3}\right)+\frac{1}{4}-\frac{1}{2}\cos (2x+\frac{\mathrm{\pi }}{3}) \\ =\frac{5}{4} \\ is a cons\tan t function \end{gathered}$$