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Understanding the Sum and Difference Formulas is essential for solving a variety of trigonometric problems. These formulas are tools that allow us to express the sine and cosine of a sum or difference of angles in terms of the sine and cosine of each individual angle. Specifically, for sine, the formulas are:

• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

When applied to an equation, these identities help to rewrite trigonometric expressions in an equivalent form that can be more easily analyzed and simplified. For example, in our exercise, we use the Sum Formula to expand \( \sin \left(\frac{\pi}{6}+x\right) \) and the Difference Formula for \( \sin \left(\frac{\pi}{6}-x\right) \). By doing so, the expression is transformed into a form where common trigonometric values can be applied and terms can be canceled to simplify the problem further.

Simplifying trigonometric expressions is a frequent task in mathematics, which involves reducing expressions to their simplest form using algebraic manipulations and trigonometric identities. This process usually requires the combination of like terms, the use of reciprocal, quotient, or Pythagorean identities, and factoring when possible. In our example, after applying the sum and difference identities, we identify and cancel out the terms that are additive inverses of each other.
For simplification, observe the following steps:

• Apply relevant trigonometric identities to expand the expressions.
• Combine like terms and cancel terms where possible.
• Factor out common factors if applicable.

In the given exercise, after applying the sum and difference formulas, we notice that the \( \cos\frac{\pi}{6}\sin{x} \) term is present with both a positive and negative sign, leading them to cancel each other. This leaves us with a much simpler expression that can be further simplified using known trigonometric values.

The values of trigonometric functions for certain standard angles are fundamental in solving trigonometric problems without a calculator. These angles usually include 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \) radians (or their degree equivalents: 0°, 30°, 45°, 60°, and 90°). Knowing these values:

• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
• \( \cos \frac{\pi}{3} = \frac{1}{2} \)

In the exercise, we use the value of \( \sin \frac{\pi}{6} \) to transform the expression into \( 2*\frac{1}{2}\cos{x} \), which simplifies down to just \( \cos{x} \). Recognizing and applying these known values is the final step that completes the process of verifying the trigonometric identity.