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 trigonometric identity for \( \tan(a + b) \).

### Step-by-Step Solution:

1. **Recall the formula for \( \tan(a + b) \)**:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Here, we will use this formula to express \( \tan 5x \).

2. **Express \( \tan 5x \) in terms of \( \tan 3x \) and \( \tan 2x \)**:
- Let \( a = 3x \) and \( b = 2x \). Then:
\[ \tan 5x = \tan(3x + 2x) = \frac{\tan 3x + \tan 2x}{1 - \tan 3x \tan 2x} \]

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

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

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

6. **Now, analyze the expression**:
- We see that the expression is quite complex, but we can conclude that it does not simplify to a specific value without specific values for \( x \).

7. **Check the options**:
- Since we derived an expression that does not match any of the options given, we conclude that the answer is **none of these**.

### Final Answer:
The value of \( \tan 5x \tan 3x \tan 2x \) does not equal any of the provided options, so the correct choice is **none of these**.

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