tan 5x tan3x `tan2x=`

Options

A. `tan5x-tan3xtan2x`
B. `(sin5x-sin3x-sin2x)/(cos5x-cos3x-cos2x)`
C. 0
D. none of these

Answer

To solve the equation \( \tan 5x \tan 3x \tan 2x \), we can use the tangent addition formula. The tangent addition formula states:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

### Step-by-Step Solution:

1. **Identify the angles**: We have \( 5x \), \( 3x \), and \( 2x \). We can express \( \tan 5x \) in terms of \( \tan 2x \) and \( \tan 3x \).

2. **Apply the tangent addition formula**:
\[ \tan 5x = \tan(2x + 3x) = \frac{\tan 2x + \tan 3x}{1 - \tan 2x \tan 3x} \]

3. **Substitute into the original expression**:
\[ \tan 5x \tan 3x \tan 2x = \left(\frac{\tan 2x + \tan 3x}{1 - \tan 2x \tan 3x}\right) \tan 3x \tan 2x \]

4. **Multiply through**:
\[ = \frac{(\tan 2x + \tan 3x) \tan 3x \tan 2x}{1 - \tan 2x \tan 3x} \]

5. **Simplify the numerator**:
\[ = \frac{\tan 2x \tan 3x + \tan^2 3x \tan 2x}{1 - \tan 2x \tan 3x} \]

6. **Rearranging**: We can also express \( \tan 3x \) in terms of \( \tan 2x \) and \( \tan x \) if needed, but we will keep it in this form for now.

7. **Final expression**: The expression simplifies to:
\[ \tan 5x \tan 3x \tan 2x = \frac{\tan 2x + \tan 3x}{1 - \tan 2x \tan 3x} \tan 3x \tan 2x \]

8. **Conclusion**: After checking the options, we find that none of the options match our derived expression.

### Final Answer:
None of these.