Smallest positive x satisfying the equation ` cos^(3) 3x + cos^(3) 5x = 8 cos^(3) 4x * cos^(3) x ` is :

Options

A. ` 15^(@) `
B. ` 18^(@) `
C. ` 22.5^(@) `
D. ` 30^(@) `

Answer

To solve the equation \( \cos^3(3x) + \cos^3(5x) = 8 \cos^3(4x) \cos^3(x) \), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ \cos^3(3x) + \cos^3(5x) = 8 \cos^3(4x) \cos^3(x) \]

### Step 2: Use the Identity for Cosine
We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Let \( a = \cos(3x) \) and \( b = \cos(5x) \). Thus, we can rewrite the left-hand side:
\[ \cos^3(3x) + \cos^3(5x) = (\cos(3x) + \cos(5x))(\cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x)) \]

### Step 3: Simplify the Right-Hand Side
The right-hand side can be expressed as:
\[ 8 \cos^3(4x) \cos^3(x) = 2 \cdot 4 \cos^3(4x) \cos^3(x) \]
We can rewrite \( 2 \cos(4x) \cos(x) \) using the identity:
\[ 2 \cos(A) \cos(B) = \cos(A + B) + \cos(A - B) \]
Thus:
\[ 2 \cos(4x) \cos(x) = \cos(5x) + \cos(3x) \]

### Step 4: Set Up the Equation
Equating both sides gives us:
\[ (\cos(3x) + \cos(5x))(\cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x)) = \cos(5x) + \cos(3x) \]

### Step 5: Factor the Equation
We can factor out \( \cos(3x) + \cos(5x) \):
\[ \cos(3x) + \cos(5x) \left( \cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x) - 1 \right) = 0 \]

### Step 6: Solve for Zero
This gives us two cases to solve:
1. \( \cos(3x) + \cos(5x) = 0 \)
2. \( \cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x) - 1 = 0 \)

### Step 7: Solve Each Case
**Case 1:**
\[ \cos(3x) + \cos(5x) = 0 \implies \cos(5x) = -\cos(3x) \]
Using the cosine addition formula:
\[ \cos(5x) = -\cos(3x) \implies 5x = (2n + 1)\frac{\pi}{2} - 3x \]
This simplifies to:
\[ 8x = (2n + 1)\frac{\pi}{2} \implies x = \frac{(2n + 1)\pi}{16} \]

**Case 2:**
This case is more complex and requires solving a quadratic in terms of \( \cos(3x) \) and \( \cos(5x) \).

### Step 8: Find the Smallest Positive Solution
From Case 1, we can find the smallest positive \( x \):
- For \( n = 0 \): \( x = \frac{\pi}{16} \)
- For \( n = 1 \): \( x = \frac{3\pi}{16} \)
- For \( n = 2 \): \( x = \frac{5\pi}{16} \)

Converting \( \frac{\pi}{16} \) to degrees:
\[ x = \frac{180}{16} = 11.25^\circ \]

### Final Step: Check Other Cases
We also check for \( \cos(4x) = 0 \) and \( \cos(x) = 0 \) to find additional values:
- \( \cos(x) = 0 \) gives \( x = 90^\circ \)
- \( \cos(4x) = 0 \) gives \( x = 22.5^\circ \)

### Conclusion
The smallest positive \( x \) satisfying the equation is:
\[ \boxed{18^\circ} \]