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The double-angle formulas are a set of trigonometric identities that express trigonometric functions of double angles, such as \(2x\), \(2\theta\), in terms of trigonometric functions of single angles, such as \(x\) or \(\theta\). In particular, the double-angle formula for cosine states that:

\[ \cos(2\theta) = 2 \cos^2(\theta) - 1 \]
This can also be written using sine:\[ \cos(2\theta) = 1 - 2 \sin^2(\theta) \]
or mixing both sine and cosine:\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \]
These formulas are extremely useful for simplifying expressions involving trigonometric functions and are vital when solving trigonometric equations or proving other identities.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. These identities are useful for simplifying complex trigonometric expressions and solving trigonometric equations. Some fundamental trigonometric identities include the Pythagorean identities:

\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
\[ \tan^2(\theta) + 1 = \sec^2(\theta) \]
\[ 1 + \cot^2(\theta) = \csc^2(\theta) \]
These Pythagorean identities highlight the inherent relationships between the trigonometric functions. Knowing these identities is critical when working with any kind of trigonometric problems, as they allow the transformation of one trigonometric function into another and play a key role in simplifications, integrations, and solving.

Simplifying a cosine function often involves reducing a complex trigonometric expression into a simpler form by using trigonometric identities. In the context of the given exercise, where we derive \(\cos(4x)\) in terms of \(\cos(x)\), the process involves using the double-angle formula multiple times. Starting with \(\cos(4x)\), we progress in steps, where in each step, we halve the angle and apply the double-angle formula. This, coupled with algebraic simplification, leads us to a more manageable expression for \(\cos(4x)\):

\[ \cos(4x) = 8 \cos^4(x) - 8 \cos^2(x) + 1 \]
This expression is considerably simpler because it uses only a single trigonometric function, \(\cos(x)\), and is composed of algebraic powers, making it much easier to integrate, differentiate, or use in further trigonometric equations.