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 equation:
\[ \cos^3(3x) + \cos^3(5x) = 8 \cos^3(4x) \cos^3(x) \]

### Step 2: Use the Sum of Cubes Formula
Recall the formula for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Let \( a = \cos(3x) \) and \( b = \cos(5x) \). Then 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 rewritten as:
\[ 8 \cos^3(4x) \cos^3(x) = 2(\cos(4x) \cos(x))^3 \]

### Step 4: Use the Cosine Addition Formula
Using the cosine addition formula:
\[ \cos(a) + \cos(b) = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]
We apply this to \( \cos(3x) + \cos(5x) \):
\[ \cos(3x) + \cos(5x) = 2 \cos(4x) \cos(x) \]

### Step 5: Substitute Back into the Equation
Substituting back, we have:
\[ 2 \cos(4x) \cos(x) \cdot (\cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x)) = 2(\cos(4x) \cos(x))^3 \]

### Step 6: Cancel Out Common Factors
Assuming \( \cos(4x) \cos(x) \neq 0 \), we can divide both sides by \( 2 \cos(4x) \cos(x) \):
\[ \cos^2(3x) - \cos(3x)\cos(5x) + \cos^2(5x) = (\cos(4x) \cos(x))^2 \]

### Step 7: Set Up the Conditions
Now we have:
\[ \cos(3x) = 0 \quad \text{or} \quad \cos(5x) = 0 \quad \text{or} \quad \cos(4x) = 0 \quad \text{or} \quad \cos(x) = 0 \]

### Step 8: Solve Each Case
1. **For \( \cos(3x) = 0 \)**:
\[ 3x = \frac{\pi}{2} + n\pi \implies x = \frac{\pi}{6} + \frac{n\pi}{3} \]

2. **For \( \cos(5x) = 0 \)**:
\[ 5x = \frac{\pi}{2} + n\pi \implies x = \frac{\pi}{10} + \frac{n\pi}{5} \]

3. **For \( \cos(4x) = 0 \)**:
\[ 4x = \frac{\pi}{2} + n\pi \implies x = \frac{\pi}{8} + \frac{n\pi}{4} \]

4. **For \( \cos(x) = 0 \)**:
\[ x = \frac{\pi}{2} + n\pi \]

### Step 9: Find the Smallest Positive Solution
Now we need to find the smallest positive \( x \) from the solutions obtained:
- From \( \cos(3x) = 0 \): \( x = \frac{\pi}{6} \)
- From \( \cos(5x) = 0 \): \( x = \frac{\pi}{10} \)
- From \( \cos(4x) = 0 \): \( x = \frac{\pi}{8} \)
- From \( \cos(x) = 0 \): \( x = \frac{\pi}{2} \)

The smallest positive \( x \) is \( \frac{\pi}{10} \).

### Step 10: Convert to Degrees
To convert \( x = \frac{\pi}{10} \) to degrees:
\[ x = \frac{\pi}{10} \cdot \frac{180}{\pi} = 18^\circ \]

### Final Answer
Thus, the smallest positive \( x \) satisfying the equation is:
\[ \boxed{\frac{\pi}{10}} \text{ or } 18^\circ \]