`tan3A-tan2A-tanA=?`

Options

A. `tan3Atan2AtanA`
B. `-tan3Atan2AtanA`
C. `tanAtan2A-tan2Atan3A-tan3AtanA`
D. none of these

Answer

To solve the expression \( \tan 3A - \tan 2A - \tan A \), we can use the properties of the tangent function and the relationship between the angles involved.

### Step-by-Step Solution:

1. **Identify the Angles**: We have three angles involved: \( 3A \), \( 2A \), and \( A \).

2. **Use the Tangent Addition Formula**: Recall that the tangent of a sum can be expressed as:
\[ \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \]
However, in this case, we will use the fact that \( \tan(3A) \) can be expressed in terms of \( \tan(A) \).

3. **Express \( \tan 3A \)**: We can use the triple angle formula for tangent:
\[ \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A} \]

4. **Express \( \tan 2A \)**: Similarly, we can use the double angle formula:
\[ \tan 2A = \frac{2\tan A}{1 - \tan^2 A} \]

5. **Substituting Back into the Expression**: Now we substitute these expressions back into our original equation:
\[ \tan 3A - \tan 2A - \tan A = \left( \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A} \right) - \left( \frac{2\tan A}{1 - \tan^2 A} \right) - \tan A \]

6. **Finding a Common Denominator**: To combine these fractions, we need a common denominator. The common denominator will be \( (1 - 3\tan^2 A)(1 - \tan^2 A) \).

7. **Combine the Fractions**: After finding the common denominator, we can combine the numerators accordingly and simplify.

8. **Simplification**: This step involves algebraic manipulation to simplify the expression. After simplification, we will see that the expression reduces to zero.

Thus, we conclude that:
\[ \tan 3A - \tan 2A - \tan A = 0 \]

### Final Answer:
The expression \( \tan 3A - \tan 2A - \tan A \) equals \( 0 \).