91Ó°ÊÓ

Triple-angle formulas are powerful tools in trigonometry that allow us to express trigonometric functions involving angles that are three times another angle in terms of functions of the original angle. These formulas are derived from angle addition identities and are particularly helpful in simplifying complex trigonometric expressions.

Here are what the triple-angle formulas for sine and cosine look like:

• The triple-angle formula for sine is: \( \sin 3x = 3 \sin x - 4 \sin^3 x \)
• The triple-angle formula for cosine is: \( \cos 3x = 4 \cos^3 x - 3 \cos x \)

Using these formulas allows us to rewrite trigonometric expressions in a form that is often easier to work with, especially in proving identities or solving equations. In the original solution, these formulas help break down \( \sin 3x \) and \( \cos 3x \) into terms that we can manipulate further to reach the desired identity.

The Pythagorean identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine functions. It states that for any angle \( x \), the following equation holds true:

\[ \sin^2 x + \cos^2 x = 1 \]
This identity is incredibly useful for transforming expressions and simplifying trigonometric equations. By rearranging this identity, we can express \( \sin^2 x \) as \( 1 - \cos^2 x \).

In our exercise, we substitute \( \sin^2 x = 1 - \cos^2 x \) to simplify the numerator in the expression. Notice how this substitution allows us to combine like terms and ultimately simplify the expression further. The Pythagorean identity serves as a bridge making complex expressions more manageable.

Cotangent is one of the six fundamental trigonometric functions, defined as the reciprocal of the tangent function. In terms of sine and cosine, cotangent of an angle \( x \) is expressed as:

\[ \cot x = \frac{\cos x}{\sin x} \]
This identity is particularly useful when dealing with expressions that can be simplified by factoring or manipulating sine and cosine terms.

In the exercise at hand, we conclude by utilizing the identity for cotangent to reach the solution. After simplifying the expression through various steps, we adopt this identity to transform the expression into a recognizable form \(-\cot 2x\).

Working with cotangent often requires familiarity with manipulating fractions and understanding reciprocal relationships between trigonometric functions. This understanding is crucial for solving trigonometric equations and verifying identities, as demonstrated in the final step of our solution.