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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.

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** 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.