If `K^(@)` lies between `360^(@) and 540^(@) and K^(@)` satisfies the equation
`1+cos10x cos6x =2cos^(2)8x+sin^(2)8x," then "(K)/(10)=`

Answer

To solve the equation \( 1 + \cos(10x) \cos(6x) = 2 \cos^2(8x) + \sin^2(8x) \) and find the value of \( \frac{K}{10} \) where \( K \) lies between \( 360^\circ \) and \( 540^\circ \), we will follow these steps:

### Step 1: Simplify the Left Side
We start with the left side of the equation:
\[ 1 + \cos(10x) \cos(6x) \]
Using the product-to-sum identities, we know:
\[ \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) \]
So, we can rewrite \( \cos(10x) \cos(6x) \):
\[ \cos(10x) \cos(6x) = \frac{1}{2} (\cos(16x) + \cos(4x)) \]
Thus, the left side becomes:
\[ 1 + \frac{1}{2} (\cos(16x) + \cos(4x)) = \frac{2}{2} + \frac{1}{2} (\cos(16x) + \cos(4x)) = \frac{2 + \cos(16x) + \cos(4x)}{2} \]

### Step 2: Simplify the Right Side
Now, we simplify the right side:
\[ 2 \cos^2(8x) + \sin^2(8x) \]
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ 2 \cos^2(8x) + (1 - \cos^2(8x)) = 2 \cos^2(8x) + 1 - \cos^2(8x) = \cos^2(8x) + 1 \]
We can express \( \cos^2(8x) \) in terms of \( \cos(16x) \):
\[ \cos^2(8x) = \frac{1 + \cos(16x)}{2} \]
Thus, the right side becomes:
\[ \cos^2(8x) + 1 = \frac{1 + \cos(16x)}{2} + 1 = \frac{1 + \cos(16x) + 2}{2} = \frac{3 + \cos(16x)}{2} \]

### Step 3: Set the Two Sides Equal
Now we have:
\[ \frac{2 + \cos(16x) + \cos(4x)}{2} = \frac{3 + \cos(16x)}{2} \]
Multiplying both sides by 2 to eliminate the denominator:
\[ 2 + \cos(16x) + \cos(4x) = 3 + \cos(16x) \]
Subtracting \( \cos(16x) \) from both sides:
\[ 2 + \cos(4x) = 3 \]
Thus, we simplify to:
\[ \cos(4x) = 1 \]

### Step 4: Solve for \( x \)
The general solution for \( \cos \theta = 1 \) is:
\[ \theta = 2n\pi \]
So we have:
\[ 4x = 2n\pi \implies x = \frac{n\pi}{2} \]

### Step 5: Find \( K \)
We need to find \( K \) such that \( K \) lies between \( 360^\circ \) and \( 540^\circ \). We convert \( x \) to degrees:
\[ x = \frac{n \cdot 180^\circ}{2} = 90n^\circ \]
Now we check for values of \( n \):
- For \( n = 4 \): \( x = 360^\circ \)
- For \( n = 5 \): \( x = 450^\circ \)

Since \( K \) must be between \( 360^\circ \) and \( 540^\circ \), we take \( K = 450^\circ \).

### Step 6: Calculate \( \frac{K}{10} \)
Finally, we calculate:
\[ \frac{K}{10} = \frac{450}{10} = 45 \]

### Final Answer
Thus, the value of \( \frac{K}{10} \) is:
\[ \boxed{45} \]