The value of `tan 3A - tan 2A - tan A` is equal to

Options

A. tan 3A tan 2A tan A
B. `-`tan 3A tan 2A tan A
C. tanA tan 2A - tan2A tan 3A - tan 3A tan A
D. none of these

Answer

To find the value of \( \tan 3A - \tan 2A - \tan A \), we can use the tangent addition formula. Here’s a step-by-step solution:

### Step 1: Express \( \tan 3A \)
Using the tangent addition formula, we can express \( \tan 3A \) as:
\[ \tan 3A = \tan(2A + A) = \frac{\tan 2A + \tan A}{1 - \tan 2A \tan A} \]

### Step 2: Substitute \( \tan 3A \) into the expression
Now, substitute this expression for \( \tan 3A \) into \( \tan 3A - \tan 2A - \tan A \):
\[ \tan 3A - \tan 2A - \tan A = \frac{\tan 2A + \tan A}{1 - \tan 2A \tan A} - \tan 2A - \tan A \]

### Step 3: Combine the terms
To combine the terms, we need a common denominator:
\[ = \frac{\tan 2A + \tan A - \tan 2A(1 - \tan 2A \tan A) - \tan A(1 - \tan 2A \tan A)}{1 - \tan 2A \tan A} \]

### Step 4: Simplify the numerator
Now, simplify the numerator:
\[ = \frac{\tan 2A + \tan A - \tan 2A + \tan 2A \tan^2 A - \tan A + \tan A \tan 2A}{1 - \tan 2A \tan A} \]
This simplifies to:
\[ = \frac{\tan 2A \tan^2 A + \tan 2A \tan A}{1 - \tan 2A \tan A} \]

### Step 5: Factor out common terms
Factor out \( \tan 2A \):
\[ = \frac{\tan 2A (\tan^2 A + \tan A)}{1 - \tan 2A \tan A} \]

### Step 6: Final expression
Thus, the final expression for \( \tan 3A - \tan 2A - \tan A \) is:
\[ = \frac{\tan 2A (\tan^2 A + \tan A)}{1 - \tan 2A \tan A} \]

### Conclusion
The value of \( \tan 3A - \tan 2A - \tan A \) is given by the expression derived above.

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