Solve : `cos 3x. Cos^(3)x+sin 3x.sin^(3)x=0`

Explanation

To solve the equation \( \cos 3x \cdot \cos^3 x + \sin 3x \cdot \sin^3 x = 0 \), we can follow these steps:

### Step 1: Rewrite \( \cos 3x \) and \( \sin 3x \)

Using the trigonometric identities:
\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]
\[ \sin 3x = 3 \sin x - 4 \sin^3 x \]

Substituting these into the equation gives:
\[ (4 \cos^3 x - 3 \cos x) \cdot \cos^3 x + (3 \sin x - 4 \sin^3 x) \cdot \sin^3 x = 0 \]

### Step 2: Expand the equation

Expanding the equation:
\[ (4 \cos^6 x - 3 \cos^4 x) + (3 \sin^4 x - 4 \sin^6 x) = 0 \]

### Step 3: Combine like terms

Rearranging gives:
\[ 4 \cos^6 x - 4 \sin^6 x - 3 \cos^4 x + 3 \sin^4 x = 0 \]

### Step 4: Factor the equation

We can factor out the common terms:
\[ 4 (\cos^6 x - \sin^6 x) - 3 (\cos^4 x - \sin^4 x) = 0 \]

Using the identity \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \) and \( a^2 - b^2 = (a-b)(a+b) \):
\[ 4 (\cos^2 x - \sin^2 x)(\cos^4 x + \cos^2 x \sin^2 x + \sin^4 x) - 3 (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) = 0 \]

### Step 5: Factor out \( (\cos^2 x - \sin^2 x) \)

Factoring out \( (\cos^2 x - \sin^2 x) \):
\[ (\cos^2 x - \sin^2 x) \left( 4 (\cos^4 x + \cos^2 x \sin^2 x + \sin^4 x) - 3 \right) = 0 \]

### Step 6: Solve the factors

1. **First Factor**:
\[ \cos^2 x - \sin^2 x = 0 \implies \cos^2 x = \sin^2 x \implies \tan^2 x = 1 \implies \tan x = \pm 1 \]
This gives:
\[ x = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]

2. **Second Factor**:
\[ 4 (\cos^4 x + \cos^2 x \sin^2 x + \sin^4 x) - 3 = 0 \]
This simplifies to:
\[ \cos^4 x + \sin^4 x + \frac{1}{4} = \frac{3}{4} \implies \cos^4 x + \sin^4 x = \frac{1}{2} \]

Using the identity \( \cos^4 x + \sin^4 x = (\cos^2 x + \sin^2 x)^2 - 2 \cos^2 x \sin^2 x \):
\[ 1 - 2 \cos^2 x \sin^2 x = \frac{1}{2} \implies 2 \cos^2 x \sin^2 x = \frac{1}{2} \implies \cos^2 x \sin^2 x = \frac{1}{4} \]
This gives:
\[ \sin^2 2x = 1 \implies 2x = \frac{\pi}{2} + n\pi \implies x = \frac{\pi}{4} + \frac{n\pi}{2} \]

### Final Solution

Combining both results, we have:
\[ x = \frac{\pi}{4} + n\pi \quad \text{and} \quad x = \frac{\pi}{4} + \frac{n\pi}{2} \]