Find the maximum value of $f(x)={\sin }^2(x)+{\sin }^2(x+\frac{\pi }{3})+\cos (x)\cos (x+\frac{\pi }{3})$.

Solution

1. Expand the terms using trigonometric identities: ${\sin }^2(x)+{\sin }^2(x+\pi /3)+\cos (x)\cos (x+\pi /3)$.
2. Use ${\sin }^2(A)=\frac{1-\cos (2A)}{2}$ and $\cos (A)\cos (B)=\frac{\cos (A-B)+\cos (A+B)}{2}$.
3. The expression becomes $\frac{1-\cos (2x)}{2}+\frac{1-\cos (2x+2\pi /3)}{2}+\frac{\cos (\pi /3)+\cos (2x+\pi /3)}{2}$.
4. Summing these terms results in $1-\frac{1}{2}(\cos (2x)+\cos (2x+2\pi /3)-\cos (2x+\pi /3))+1/4=3/2$.
5. Hence the answer is (B).