The number of values of x lying in the inteval `(-2pi, 2pi)` satisfying the equation `1+cos 10x cos 6x=2 cos^(2)8x+sin^(2)8x` is equal to

Answer

To solve the equation \(1 + \cos(10x) \cos(6x) = 2 \cos^2(8x) + \sin^2(8x)\) for the number of values of \(x\) in the interval \((-2\pi, 2\pi)\), we can follow these steps:

### Step 1: Simplify the Right Side
We know that \(2 \cos^2(8x) + \sin^2(8x)\) can be rewritten using the Pythagorean identity:
\[ \sin^2(8x) = 1 - \cos^2(8x) \]
Thus, we have:
\[ 2 \cos^2(8x) + \sin^2(8x) = 2 \cos^2(8x) + (1 - \cos^2(8x)) = \cos^2(8x) + 1 \]

### Step 2: Rewrite the Equation
Now, substituting this back into the equation gives us:
\[ 1 + \cos(10x) \cos(6x) = \cos^2(8x) + 1 \]
We can simplify this by subtracting 1 from both sides:
\[ \cos(10x) \cos(6x) = \cos^2(8x) \]

### Step 3: Use the Product-to-Sum Formula
Using the product-to-sum formulas, we can express \(\cos(10x) \cos(6x)\) as:
\[ \cos(10x) \cos(6x) = \frac{1}{2} [\cos(10x - 6x) + \cos(10x + 6x)] = \frac{1}{2} [\cos(4x) + \cos(16x)] \]
Thus, our equation becomes:
\[ \frac{1}{2} [\cos(4x) + \cos(16x)] = \cos^2(8x) \]

### Step 4: Multiply Through by 2
Multiplying both sides by 2 gives:
\[ \cos(4x) + \cos(16x) = 2 \cos^2(8x) \]

### Step 5: Use the Double Angle Identity
The right side can be rewritten using the double angle identity:
\[ 2 \cos^2(8x) = 1 + \cos(16x) \]
Substituting this into our equation gives:
\[ \cos(4x) + \cos(16x) = 1 + \cos(16x) \]
Subtracting \(\cos(16x)\) from both sides results in:
\[ \cos(4x) = 1 \]

### Step 6: Solve for \(x\)
The equation \(\cos(4x) = 1\) implies:
\[ 4x = 2n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus, we have:
\[ x = \frac{n\pi}{2} \]

### Step 7: Determine Values of \(n\)
Now we need to find the integer values of \(n\) such that \(x\) lies in the interval \((-2\pi, 2\pi)\):
\[ -2\pi < \frac{n\pi}{2} < 2\pi \]
Multiplying through by 2 gives:
\[ -4 < n < 4 \]
The integer values of \(n\) that satisfy this inequality are:
\(-3, -2, -1, 0, 1, 2, 3\)

### Step 8: Count the Values
Counting these values gives us a total of 7 values of \(x\) in the interval \((-2\pi, 2\pi)\).

### Final Answer
The number of values of \(x\) lying in the interval \((-2\pi, 2\pi)\) satisfying the equation is **7**.

---