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The sine difference formula is a powerful identity in trigonometry that assists in simplifying expressions involving the difference of two sine angles. The formula is expressed as:

• \( \sin A - \sin B = 2 \cos\left( \frac{A+B}{2} \right) \sin\left( \frac{A-B}{2} \right) \)

This formula works by transforming the difference of two sine functions into a product of a cosine and a sine function. This is often easier to manage and further simplifies the expression. In our exercise involving \( \sin(10x) \) and \( \sin(2x) \), substituting them into the sine difference formula simplifies the calculation. By doing this, we obtain \( 2 \cos(6x) \sin(4x) \), making the expression more manageable as we move closer to proving the identity.

The cosine sum formula provides a means to simplify expressions that involve adding two cosine functions. It is stated as:

• \( \cos A + \cos B = 2 \cos\left( \frac{A+B}{2} \right) \cos\left( \frac{A-B}{2} \right) \)

Using the cosine sum formula swaps a sum of two cosine terms for a product of two cosine terms, which can be profoundly helpful in further simplifications. In the given problem, \( \cos(10x) + \cos(2x) \) is transformed into the product \( 2 \cos(6x) \cos(4x) \), thereby simplifying the computation significantly. This transformation facilitates the eventual simplification needed to prove that the original expression simplifies to \( \tan(4x) \).

Simplifying expressions is a key skill in solving trigonometric identities and equations. It involves reducing a complex expression to its simplest form for easier computation and understanding. In the context of trigonometric identities like the one given, simplification is often achieved

• Using known trigonometric identities such as sine and cosine formulas.
• Canceling common terms in numerators and denominators.
• Rewriting expressions in terms of basic trigonometric functions like \( \tan, \sin, \cos \).

In this problem, after utilizing the sine difference and cosine sum formulas, we arrive at an expression \( \frac{2\cos(6x)\sin(4x)}{2\cos(6x)\cos(4x)} \). By cancelling the common factor, \( 2\cos(6x) \), in both the numerator and denominator, we simplify the expression to \( \tan(4x) \). This step crucially helps in verifying the identity by reducing the expression to its simplest form, ultimately matching the target outcome.