The number of solutions of the equation `cos^2(x+pi/6)+cos^2x-2cos(x+pi/6)dotcospi/6=sin^2pi/6` in interval `((-pi)/2,pi/2)` is_________

Explanation

To solve the equation \( \cos^2\left(x + \frac{\pi}{6}\right) + \cos^2 x - 2 \cos\left(x + \frac{\pi}{6}\right) \cdot \cos\left(\frac{\pi}{6}\right) = \sin^2\left(\frac{\pi}{6}\right) \) in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we can follow these steps:

### Step 1: Simplify the equation
We know that \( \sin^2\left(\frac{\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) and \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \). Thus, we can rewrite the equation as:

\[ \cos^2\left(x + \frac{\pi}{6}\right) + \cos^2 x - 2 \cdot \cos\left(x + \frac{\pi}{6}\right) \cdot \frac{\sqrt{3}}{2} = \frac{1}{4} \]

### Step 2: Use the cosine addition formula
Using the cosine addition formula, we have:

\[ \cos\left(x + \frac{\pi}{6}\right) = \cos x \cos\left(\frac{\pi}{6}\right) - \sin x \sin\left(\frac{\pi}{6}\right) \]

Substituting \( \cos\left(\frac{\pi}{6}\right) \) and \( \sin\left(\frac{\pi}{6}\right) \):

\[ \cos\left(x + \frac{\pi}{6}\right) = \cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2} \]

### Step 3: Substitute back into the equation
Now, substituting this back into the equation gives us:

\[ \cos^2\left(x + \frac{\pi}{6}\right) + \cos^2 x - 2 \cdot \left(\cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2}\right) \cdot \frac{\sqrt{3}}{2} = \frac{1}{4} \]

### Step 4: Expand and simplify
Expanding \( \cos^2\left(x + \frac{\pi}{6}\right) \) using the square of the cosine addition formula will be complex. Instead, we can use the identity \( a^2 + b^2 - 2ab = (a - b)^2 \) where \( a = \cos\left(x + \frac{\pi}{6}\right) \) and \( b = \cos\left(\frac{\pi}{6}\right) \).

Thus, we can rewrite the left-hand side as:

\[ \left(\cos\left(x + \frac{\pi}{6}\right) - \cos\left(\frac{\pi}{6}\right)\right)^2 + \cos^2 x = \frac{1}{4} \]

### Step 5: Set up the equation for solutions
This leads us to find the values of \( x \) that satisfy:

\[ \left(\cos\left(x + \frac{\pi}{6}\right) - \frac{\sqrt{3}}{2}\right)^2 + \cos^2 x = \frac{1}{4} \]

### Step 6: Solve for \( x \)
Now, we will analyze the solutions. The equation can be solved by finding where the left-hand side equals \( \frac{1}{4} \).

1. **First solution**: \( x = 0 \)
2. **Second solution**: \( x = \frac{\pi}{3} \)

### Step 7: Count the solutions
We need to check if these solutions lie within the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Both \( x = 0 \) and \( x = \frac{\pi}{3} \) are valid solutions.

### Final Answer
Thus, the number of solutions of the equation in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) is **2**.

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