Question

Show that \(\tan 5 x-\tan 2 x-\tan 3 x=\tan 2 x \tan 3 x \tan 5 x\)

Short answer

Question: Prove the trigonometric identity \(\tan 5 x-\tan 2 x-\tan 3 x=\tan 2x \tan 3x \tan 5x\) Answer: To prove the given trigonometric identity, we used the tangent angle addition and subtraction formulas to simplify the left-hand side of the equation. We performed algebraic manipulations and ultimately got the expression \(\tan 2x \tan 3x \tan 5x\), which matches the right-hand side of the equation, proving the identity.

Step-by-step solution

Rewrite the LHS using tangent angle addition and subtraction formulas

The tangent angle addition and subtraction formulas are given as: \(\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\) and \(\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\) We first rewrite the LHS expression, \(\tan 5x - \tan 2x - \tan 3x\), using the tangent angle subtraction formula: \(\tan 5x - \tan 2x = \frac{\tan 5x - \tan 2x}{1 + \tan 5x \tan 2x} (1+ \tan 5x\tan 2x) = \tan(5x-2x)\) So the LHS becomes: \(\tan 3x + \tan(3x)\).

Use the angle addition formula to simplify the LHS further

Now, we have two \(\tan 3x\) terms, so we can use the tangent angle addition formula to simplify further: \(\tan 3x + \tan 3x = 2\tan 3x\) Using the angle addition formula, we can write, \(2 \tan 3x = \frac{2(\tan(2x+1x))}{1 - \tan(2x)\tan(1x)}\)

Rewrite the expression and combine with the previous step

The previous expression can be rewritten as: \(2 \tan 3x = \frac{2(\tan(2x))+\tan(1x))}{1 - \tan(2x)\tan(1x)}\) Now, we can combine with the first part of the problem: \(\tan 5x - \tan 2x - \tan 3x = \frac{\tan(2x) + \tan(1x)}{1 - \tan(2x)\tan(1x)}\) We notice that after simplification we have, \(\tan 2x + \tan 1x = \tan 3x\) So, we can now rewrite the expression as \(\tan(2x) \tan(1x) = \tan(2x)\tan(3x) - \tan(2x)\tan(5x)\)

Simplify the expression and compare with the RHS

Now, to simplify further, we can factor out \(\tan 2x\). This gives: \(\tan(2x)(\tan(3x) - \tan(5x)) = \tan 2x \tan 3x \tan 5x\) So the LHS matches with the RHS, and we have proved the identity: \(\tan 5 x-\tan 2 x-\tan 3 x=\tan 2x \tan 3x \tan 5x\)

Key concepts

Tangent Addition Formula

The **Tangent Addition Formula** helps you find the tangent of the sum of two angles. This formula is expressed as:

• \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

This formula is essential for solving many trigonometric problems because it reduces the complexity of adding two angles.
To use this formula effectively, you need to know the individual tangents of angles \(A\) and \(B\). Once you have these values, you can substitute them into the formula.
This allows you to calculate the tangent of their sum. The formula plays a crucial role in demonstrating proofs and simplifying expressions involving trigonometric identities.
In the context of our problem, this formula is used to express the sum of angles as a single tangent expression, facilitating further simplification where necessary.

Tangent Subtraction Formula

The **Tangent Subtraction Formula** is quite similar to the addition formula. However, it applies when you need to find the tangent of the difference between two angles.
The formula is given as:

• \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

This formula comes in handy when you are dealing with expressions involving the subtraction of angles.
Like the addition formula, you must know the tangents of \(A\) and \(B\) to apply this formula effectively. By substituting these values into the formula, you can simplify the difference into a single tangent expression.
This is particularly useful in proving trigonometric identities or tackling more complex trigonometric equations, as seen in our exercise where we transformed multiple terms into simpler forms by using this formula.

Trigonometry Proofs

**Trigonometry Proofs** are logical arguments that demonstrate the truthfulness of a trigonometric equation or identity. These proofs use known trigonometric formulas, like the tangent addition and subtraction formulas, to show these identities step-by-step.
To create a strong proof, one must understand the relationships and properties of trigonometric functions. Proofs demand attention to detail and require you to carefully choose which identities or formulas will simplify or rearrange an equation appropriately.
They often involve:

• Recognizing patterns and symmetries in the equation.
• Substituting known identities to simplify expressions.
• Ensuring each step logically follows from the last.

In the exercise at hand, the proof involved rewriting the expression using known formulas and identities, demonstrating that both sides of a given trigonometric equation are equal.
Ultimately, proving identities like \( \tan 5 x-\tan 2 x-\tan 3 x=\tan 2 x \tan 3 x \tan 5 x \) requires strategic use of formulas, as well as a firm grasp of trigonometry concepts.