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

Options

A. cos 2x
B. `sin^(2)x`
C. `3/4`
D. none of these

Answer

To solve the expression \( \cos^2 x + \cos^2 \left( \frac{\pi}{3} + x \right) - \cos x \cdot \cos \left( \frac{\pi}{3} + x \right) \), we will follow these steps:

### Step 1: Rewrite the expression
We can rearrange the expression as follows:
\[ \cos^2 x + \cos^2 \left( \frac{\pi}{3} + x \right) - \cos x \cdot \cos \left( \frac{\pi}{3} + x \right) \]

### Step 2: Use the cosine addition formula
Recall the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Using this, we can expand \( \cos \left( \frac{\pi}{3} + x \right) \):
\[ \cos \left( \frac{\pi}{3} + x \right) = \cos \frac{\pi}{3} \cos x - \sin \frac{\pi}{3} \sin x \]
Substituting the known values \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \):
\[ \cos \left( \frac{\pi}{3} + x \right) = \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \]

### Step 3: Substitute back into the expression
Now substitute this back into the expression:
\[ \cos^2 x + \cos^2 \left( \frac{\pi}{3} + x \right) - \cos x \left( \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \right) \]

### Step 4: Expand \( \cos^2 \left( \frac{\pi}{3} + x \right) \)
Next, we need to calculate \( \cos^2 \left( \frac{\pi}{3} + x \right) \):
\[ \cos^2 \left( \frac{\pi}{3} + x \right) = \left( \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \right)^2 \]
Expanding this gives:
\[ = \frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x \]

### Step 5: Combine all terms
Now combine all the terms:
\[ \cos^2 x + \left( \frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x \right) - \left( \frac{1}{2} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x \right) \]

### Step 6: Simplify the expression
Combine like terms:
1. For \( \cos^2 x \):
\[ \cos^2 x + \frac{1}{4} \cos^2 x - \frac{1}{2} \cos^2 x = \frac{3}{4} \cos^2 x \]
2. For \( \sin^2 x \):
\[ \frac{3}{4} \sin^2 x \]
3. For \( \cos x \sin x \):
\[ -\frac{\sqrt{3}}{2} \cos x \sin x + \frac{\sqrt{3}}{2} \cos x \sin x = 0 \]

### Step 7: Final expression
Thus, we have:
\[ \frac{3}{4} \cos^2 x + \frac{3}{4} \sin^2 x = \frac{3}{4} (\cos^2 x + \sin^2 x) \]
Using the identity \( \cos^2 x + \sin^2 x = 1 \):
\[ = \frac{3}{4} \cdot 1 = \frac{3}{4} \]

### Final Answer
The value of the expression is \( \frac{3}{4} \).