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

Options

A. `tan 3A tan 2A tan A `
B. `- tan 3 A tan 2 A tan A `
C. `tan A tan 2 A - tan 2 A tan 3A - tan 3A tan A `
D. None of these

Answer

To solve the problem of finding the value of \( \tan 3A - \tan A \), we can use the tangent subtraction formula and some trigonometric identities. Here’s a step-by-step solution:

### Step 1: Use the angle subtraction identity
We know that \( \tan(3A) \) can be expressed in terms of \( \tan(A) \) using the identity for \( \tan(3A) \):
\[ \tan(3A) = \frac{3\tan(A) - \tan^3(A)}{1 - 3\tan^2(A)} \]

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

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

### Step 4: Final expression
Thus, we have:
\[ \tan(3A) - \tan(A) = \frac{2\tan(A)(1 + \tan^2(A))}{1 - 3\tan^2(A)} \]

### Conclusion
The value of \( \tan 3A - \tan A \) can be expressed as:
\[ \tan 3A - \tan A = \frac{2\tan A(1 + \tan^2 A)}{1 - 3\tan^2 A} \]