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Trigonometric equations can often be solved exactly under certain conditions. When confronted with such equations, the aim is to find specific angle measures that satisfy the equation within a given interval. In the context of the given problem, \( \cos(2x) - \cos(x) = 0 \), we need to find all angles x within the interval \( [0, 2\pi) \) that make the equation true.

After applying a double-angle identity and factoring, the equation is broken down into simpler parts, allowing us to consider each factor separately. This divide-and-conquer approach is beneficial as it reduces the complex problem into more manageable ones. Here, we look for exact values of x by utilizing inverse trigonometric functions, such as \( \cos^{-1} \) which are designed to retrieve the angle given the value of the cosine function. These solutions will always be within the principal range \( [0, \pi] \) for \( \cos^{-1} \), as per the properties of the inverse cosine function.

A successful strategy for solving trigonometric equations is to factor them, a method resembling the treatment of polynomial equations. Once the original equation \( \cos(2x) - \cos(x) = 0 \) is expressed using a trigonometric identity and rearranged into a quadratic form, we can use factoring to simplify finding the solutions.

Through factoring, we transform the equation \( 2\cos^2(x) - \cos(x) - 1 = 0 \) into a product of binomials \( (2\cos(x) - 1)(\cos(x) + 1) = 0 \). From the Zero Product Property, we know at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x individually. This procedure elucidates that trigonometric equations, much like algebraic ones, are solvable via logical, systematic methods.

The double-angle formulas are invaluable tools in trigonometry that relate the sine, cosine, and tangent of an angle to the sine, cosine, or tangent of its double. Specifically, for cosine, the double-angle formula is expressed as \( \cos(2x) = 2\cos^2(x) - 1 \).

The inherent beauty of these formulas is their ability to condense expressions and thus simplify the process of solving trigonometric equations. In the provided exercise, the double-angle formula transforms the equation, enabling us to rewrite \( \cos(2x) \) in terms of \( \cos(x) \) and subsequently factor the equation. This transformation is pivotal as it changes a transcendental equation into a form that is solvable by conventional algebraic means, such as factoring. Students should become familiar with these formulas to efficiently tackle trigonometric problems.