`tan 5x - tan 3x - tan 2x=`

Options

A. tan5x tan 3x tan 2x
B. cos 5x cos 3x cos 2x
C. sin 5x sin 3x sin 2x
D. tan 8x tan 3x tan 2x

Answer

To solve the equation \( \tan 5x - \tan 3x - \tan 2x = 0 \), we can use the tangent addition formula. Here’s a step-by-step solution:

### Step 1: Use the Tangent Addition Formula
We know that:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
We can express \( \tan 5x \) as \( \tan(3x + 2x) \).

### Step 2: Apply the Formula
Using the formula, we have:
\[ \tan 5x = \tan(3x + 2x) = \frac{\tan 3x + \tan 2x}{1 - \tan 3x \tan 2x} \]

### Step 3: Substitute into the Equation
Now, substituting this into our original equation:
\[ \tan 5x - \tan 3x - \tan 2x = 0 \]
becomes:
\[ \frac{\tan 3x + \tan 2x}{1 - \tan 3x \tan 2x} - \tan 3x - \tan 2x = 0 \]

### Step 4: Clear the Denominator
Multiply through by \( 1 - \tan 3x \tan 2x \) to eliminate the fraction:
\[ \tan 3x + \tan 2x - \tan 3x(1 - \tan 3x \tan 2x) - \tan 2x(1 - \tan 3x \tan 2x) = 0 \]

### Step 5: Expand the Equation
Expanding gives:
\[ \tan 3x + \tan 2x - \tan 3x + \tan 3x \tan 2x \tan 3x - \tan 2x + \tan 3x \tan 2x \tan 2x = 0 \]
This simplifies to:
\[ \tan 3x \tan 2x (\tan 3x + \tan 2x) = 0 \]

### Step 6: Factor the Equation
Thus, we have:
\[ \tan 3x \tan 2x (\tan 3x + \tan 2x) = 0 \]

### Step 7: Solve the Factors
This gives us two cases to consider:
1. \( \tan 3x = 0 \)
2. \( \tan 2x = 0 \)
3. \( \tan 3x + \tan 2x = 0 \)

### Step 8: Find Solutions
1. From \( \tan 3x = 0 \):
\[ 3x = n\pi \implies x = \frac{n\pi}{3} \]
2. From \( \tan 2x = 0 \):
\[ 2x = m\pi \implies x = \frac{m\pi}{2} \]
3. From \( \tan 3x + \tan 2x = 0 \):
\[ \tan 3x = -\tan 2x \]

### Conclusion
The solutions for \( x \) are:
\[ x = \frac{n\pi}{3}, \quad x = \frac{m\pi}{2} \quad \text{and} \quad \tan 3x = -\tan 2x \]