The period of the function $f(x) = e^{\sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cos\left(x + \frac{\pi}{3}\right)}$ is:
Step-by-Step Solution
Key Concept: A constant function has no well-defined period because it does not repeat cyclically.
Expanding $\sin^2 x + \sin^2(x + \frac{\pi}{3}) + \cos x\cos(x + \frac{\pi}{3}) = \sin^2 x + (\frac{\sin x}{2} - \frac{\sqrt{3}}{2}\cos x)^2 + \cos x(\frac{\cos x}{2} - \frac{\sqrt{3}}{2}\sin x)$. After simplification using $\sin^2 x + \cos^2 x = 1$, the expression reduces to $\frac{5}{4}$. Since $f(x)$ equals the constant $\frac{5}{4}$, it has no period.
Correct Answer: 4