The period of the function `f(x)=c^((sin^2x) +sin^2 (x+pi/3)+cosxcos(x+pi/3))` is (where `c` is constant)

Options

A. 1
B. `(pi)/(2)`
C. `pi`
D. None of these

Explanation

To find the period of the function

\[ f(x) = c^{\left(\sin^2 x + \sin^2 \left(x + \frac{\pi}{3}\right) + \cos x \cos \left(x + \frac{\pi}{3}\right)\right)} \]

we will analyze the components of the function step by step.

### Step 1: Analyze the components of the function

The function \( f(x) \) consists of three parts: \( \sin^2 x \), \( \sin^2 \left(x + \frac{\pi}{3}\right) \), and \( \cos x \cos \left(x + \frac{\pi}{3}\right) \).

### Step 2: Determine the period of \( \sin^2 x \)

The function \( \sin^2 x \) has a period of \( \pi \). This is because \( \sin^2 x = \left(\sin x\right)^2 \) and the sine function itself has a period of \( 2\pi \), but squaring it reduces the period to \( \pi \).

### Step 3: Determine the period of \( \sin^2 \left(x + \frac{\pi}{3}\right) \)

The function \( \sin^2 \left(x + \frac{\pi}{3}\right) \) also has the same period as \( \sin^2 x \), which is \( \pi \). The phase shift does not affect the period.

### Step 4: Determine the period of \( \cos x \cos \left(x + \frac{\pi}{3}\right) \)

Using the product-to-sum identities, we can rewrite \( \cos x \cos \left(x + \frac{\pi}{3}\right) \):

\[ \cos x \cos \left(x + \frac{\pi}{3}\right) = \frac{1}{2} \left(\cos(2x + \frac{\pi}{3}) + \cos \frac{\pi}{3}\right) \]

The cosine function has a period of \( 2\pi \), and thus \( \cos(2x + \frac{\pi}{3}) \) has a period of \( \pi \). Therefore, \( \cos x \cos \left(x + \frac{\pi}{3}\right) \) also has a period of \( \pi \).

### Step 5: Combine the periods

Since all three components \( \sin^2 x \), \( \sin^2 \left(x + \frac{\pi}{3}\right) \), and \( \cos x \cos \left(x + \frac{\pi}{3}\right) \) have the same period of \( \pi \), the overall function \( f(x) \) will also have a period of \( \pi \).

### Conclusion

Thus, the period of the function \( f(x) \) is:

\[ \text{Period} = \pi \]