If `f (x) = cos^2 x + cos^2 (pi/3 +x)- cos x cos (pi/3 + x)` then f (x) is :

Options

A. odd
B. even
C. periodic
D. `f(0) = f(1)`

Answer

To solve the function \( f(x) = \cos^2 x + \cos^2 \left(\frac{\pi}{3} + x\right) - \cos x \cos \left(\frac{\pi}{3} + x\right) \), we will simplify it step by step.

### Step 1: Expand \( \cos^2 \left(\frac{\pi}{3} + x\right) \)

Using the cosine addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
we can express \( \cos \left(\frac{\pi}{3} + x\right) \):
\[ \cos \left(\frac{\pi}{3} + x\right) = \cos \frac{\pi}{3} \cos x - \sin \frac{\pi}{3} \sin x \]
Substituting the known values:
\[ \cos \frac{\pi}{3} = \frac{1}{2}, \quad \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos \left(\frac{\pi}{3} + x\right) = \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \]

### Step 2: Calculate \( \cos^2 \left(\frac{\pi}{3} + x\right) \)

Now we square the expression:
\[ \cos^2 \left(\frac{\pi}{3} + x\right) = \left(\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x\right)^2 \]
Expanding this:
\[ = \left(\frac{1}{2}\right)^2 \cos^2 x - 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \cos x \sin x + \left(\frac{\sqrt{3}}{2}\right)^2 \sin^2 x \]
\[ = \frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x \]

### Step 3: Substitute into \( f(x) \)

Now substitute this back into \( f(x) \):
\[ f(x) = \cos^2 x + \left(\frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x\right) - \cos x \left(\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x\right) \]

### Step 4: Simplify \( f(x) \)

Combine like terms:
\[ = \cos^2 x + \frac{1}{4} \cos^2 x + \frac{3}{4} \sin^2 x - \frac{\sqrt{3}}{2} \cos x \sin x - \left(\frac{1}{2} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x\right) \]
\[ = \left(1 + \frac{1}{4} - \frac{1}{2}\right) \cos^2 x + \frac{3}{4} \sin^2 x - \left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\right) \cos x \sin x \]
\[ = \frac{3}{4} \cos^2 x + \frac{3}{4} \sin^2 x \]

### Step 5: Factor out the common term

Factoring out \( \frac{3}{4} \):
\[ f(x) = \frac{3}{4} (\cos^2 x + \sin^2 x) \]
Using the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \):
\[ f(x) = \frac{3}{4} \cdot 1 = \frac{3}{4} \]

### Conclusion

Thus, the function \( f(x) \) simplifies to a constant value:
\[ f(x) = \frac{3}{4} \]