The maximum value of `cos^2(pi/3-x)-cos^2(pi/3+x)`, is

Options

A. `(sqrt3)/(2)`
B. `1/2`
C. `-(sqrt3)/(2)`
D. `3/2`

Answer

To find the maximum value of the expression \( \cos^2\left(\frac{\pi}{3} - x\right) - \cos^2\left(\frac{\pi}{3} + x\right) \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \cos^2\left(\frac{\pi}{3} - x\right) - \cos^2\left(\frac{\pi}{3} + x\right) \]

### Step 2: Use the identity for the difference of squares
We can use the identity \( \cos^2 A - \cos^2 B = \sin(A - B) \cdot \sin(A + B) \). Here, let \( A = \frac{\pi}{3} - x \) and \( B = \frac{\pi}{3} + x \).

Thus, we can rewrite the expression as:
\[ \sin\left(\left(\frac{\pi}{3} - x\right) - \left(\frac{\pi}{3} + x\right)\right) \cdot \sin\left(\left(\frac{\pi}{3} - x\right) + \left(\frac{\pi}{3} + x\right)\right) \]

### Step 3: Simplify the arguments of the sine functions
Calculating the first sine term:
\[ \left(\frac{\pi}{3} - x\right) - \left(\frac{\pi}{3} + x\right) = -2x \]
Calculating the second sine term:
\[ \left(\frac{\pi}{3} - x\right) + \left(\frac{\pi}{3} + x\right) = \frac{2\pi}{3} \]

Now, substituting back, we have:
\[ \sin(-2x) \cdot \sin\left(\frac{2\pi}{3}\right) \]

### Step 4: Use the property of sine
Using the property \( \sin(-\theta) = -\sin(\theta) \):
\[ -\sin(2x) \cdot \sin\left(\frac{2\pi}{3}\right) \]

### Step 5: Calculate \( \sin\left(\frac{2\pi}{3}\right) \)
We know that:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 6: Substitute back into the expression
Now substituting this value back into our expression:
\[ -\sin(2x) \cdot \frac{\sqrt{3}}{2} \]

### Step 7: Find the maximum value
The maximum value of \( -\sin(2x) \) occurs when \( \sin(2x) \) is at its minimum, which is -1. Therefore:
\[ \text{Maximum value} = -(-1) \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

### Conclusion
Thus, the maximum value of the expression \( \cos^2\left(\frac{\pi}{3} - x\right) - \cos^2\left(\frac{\pi}{3} + x\right) \) is:
\[ \boxed{\frac{\sqrt{3}}{2}} \]