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Double angle formulas are a cornerstone concept in solving trigonometric equations. They express trigonometric functions of double angles — that is, functions like \( \text{sin}(2x) \), \( \text{cos}(2x) \), and \( \text{tan}(2x) \) — in terms of single angles. The application of these formulas is crucial when addressing trigonometric equations that involve angles' multiples.

For instance, the double angle formula for sine is \( \text{sin}(2x) = 2\text{sin}(x)\text{cos}(x) \). This formula was employed in the exercise to transform the original equation from a double angle representation to a product of single angle functions, making it easier to solve. Double angle formulas not only simplify equations but also reveal opportunities for factoring, which the next steps of the textbook solution effectively demonstrate.

Factoring is an essential algebraic tool that applies to trigonometric equations as well. When an equation contains a common trigonometric function or expression within its terms, factoring can simplify the equation, as seen in the exercise where \( \text{cos}(x) \) was the common factor.

By factoring out \( \text{cos}(x) \), we reduce the equation from a sum of terms to a product of factors. This greatly simplifies the equation, allowing us to use the Zero Product Property. This property tells us if a product of factors equals zero, then at least one of the factors must be zero. Hence, the problem dissolves into solving simpler equations for each factor individually. Factoring in trigonometry often simplifies complex relationships and is a tool that, once mastered, vastly improves a student's ability to solve trigonometric equations.

Understanding trigonometric roots is fundamental when determining the solutions of trigonometric equations within a specific interval. These roots are the angle measures that satisfy the given trigonometric equation. In the context of the exercise, setting the factored expressions equal to zero leads to the identification of the trigonometric roots.

Each trigonometric function has a set of principal angle solutions. For example, \( \text{cos}(x) = 0 \) at \( x = \frac{\text{Ï€}}{2} \) and \( x = \frac{3\text{Ï€}}{2} \) within one cycle of the unit circle, which corresponds to the interval \( [0, 2\text{Ï€}) \). When finding trigonometric roots, it is important to consider the function's periodicity and the specific interval required for the solution. Properly identifying and verifying these roots ensures accurate solutions to trigonometric equations.