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The cosine function is a primary trigonometric function, commonly denoted as \( \cos \), crucial in the field of mathematics, especially in the study of triangles and circles. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. However, the function extends well beyond right-angled triangles and is defined on the unit circle for all real numbers.

When evaluating \( \cos(x) \) at different angles, we receive values between \( -1 \) and \( 1 \) inclusive, which represent coordinates on the unit circle's x-axis. In particular, the double-angle formula \( \cos(2x) = 2\cos^2(x) - 1 \) is a vital tool, derived from the cosine of a sum formula, and allows us to connect the cosine of an angle to the cosine of its double.

Proofs in trigonometry are logical arguments demonstrating that a particular relationship holds true for all permissible values of the involved variables. A proof often involves applying trigonometric identities, algebraic manipulation, and sometimes geometric interpretation to show that one side of an equation can be transformed into the other.

In our example, we begin by equating two expressions involving double angles. Using the double-angle formula for cosine and algebraically manipulating the equation, we demonstrate that the squares of cosine values are equal. We then apply the principle that if two non-negative numbers have equal squares, their absolute values must also be equal, leading to a successful proof.

Absolute value represents the distance of a number from zero on the number line, making it always non-negative. For any real number \( x \), the absolute value, denoted as \( |x| \), satisfies the properties that \( |x| \geq 0 \) and \( |x| = |-x| \). Furthermore, \( |ab| = |a||b| \) and \( \sqrt{x^2} = |x| \).

In trigonometric contexts, these properties are essential. When dealing with trigonometric functions, we are often interested in the magnitudes of these values, irrelevant of their signs. For the case of the cosine function, the property that the square root of a number's square gives its absolute value helps us to infer magnitudes for trigonometric expressions.

Trigonometric identities are equations involving trigonometric functions that are true for all admissible angle measures. They play a fundamental role in simplifying expressions, solving equations, and proving other mathematical statements. Common identities include Pythagorean identities, angle sum and difference identities, and double-angle formulas.

The double-angle formula \( \cos(2x) = 2\cos^2(x) - 1 \) used in the exercise is one of these identities. It is derived from the angle sum identity \( \cos(x + x) \) and serves as a powerful tool in our proof. Trigonometric identities are often memorized, but understanding their derivation from the unit circle can strengthen conceptual knowledge and application skills.