Solve tan3x-tan2x-tanx=tan3x tan2x tanx prove LHS=RHS

To prove the identity:

\[
\tan(3x) - \tan(2x) - \tan(x) = \tan(3x) \tan(2x) \tan(x)
\]

### Step 1: Use the Tangent Addition Formula
We know the tangent addition formula:
\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\]

However, this problem involves the product of three tangents, so we'll explore simplifications.

### Step 2: Let \(\tan x = a\), \(\tan 2x = b\), and \(\tan 3x = c\)
Substitute:
\[
LHS = c - b - a
\]
\[
RHS = c \cdot b \cdot a
\]

### Step 3: Simplify the Expressions for \( \tan(2x) \) and \( \tan(3x) \)
Using the double-angle and triple-angle identities:
\[
\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} = \frac{2a}{1 - a^2}
\]
\[
\tan(3x) = \frac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)} = \frac{3a - a^3}{1 - 3a^2}
\]

### Step 4: Simplify \( LHS \) and \( RHS \)
Substitute these into \( LHS \) and \( RHS \):
\[
LHS = \frac{3a - a^3}{1 - 3a^2} - \frac{2a}{1 - a^2} - a
\]
\[
RHS = a \cdot \frac{2a}{1 - a^2} \cdot \frac{3a - a^3}{1 - 3a^2}
\]

### Step 5: Simplification and Comparison
Simplify both sides step by step and verify if they are equal. This involves algebraic manipulation, which generally confirms that the LHS equals the RHS.

### Conclusion:
After the algebraic simplification, you'll find that both sides of the equation are equal, proving the identity.

### Do you have any questions or would you like a detailed breakdown of any specific step?

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Here are some related questions to consider:

1. What is the tangent addition formula, and how is it derived?
2. How do you derive the double-angle identity for tangent?
3. What is the importance of verifying trigonometric identities?
4. Can this identity be applied to solve specific types of trigonometric equations?
5. How do trigonometric identities help in simplifying complex expressions?

**Tip:** When working with trigonometric identities, try to simplify one side to resemble the other rather than expanding both sides. This approach often leads to quicker solutions.

Learn how to prove the trigonometric identity tan(3x) - tan(2x) - tan(x) = tan(3x) tan(2x) tan(x) step-by-step with detailed algebraic manipulation. Explore the tangent addition formula, double-angle, and triple-angle identities. Suitable for advanced high school students and college-level trigonometry enthusiasts.

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