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The double angle formula plays a crucial role in simplifying expressions involving trigonometric and hyperbolic functions. It is a mathematical relation that allows us to express functions of double angles, such as \(2x\), in terms of single angles like \(x\). This is immensely useful in calculus, physics, and engineering to break down complex problems into more manageable parts.

For hyperbolic cosine functions, the double angle formula is given by \(\text{cosh}(2x) = 2\text{cosh}^2(x) - 1\). Understanding this formula is key to working with hyperbolic identities and verifying them, as demonstrated in the given exercise. By using the double angle formula, the complex expression of \(\text{cosh}(2x)\) is reduced to a function of \(x\), simplifying the entire problem-solving process.

Hyperbolic functions, similar to their trigonometric counterparts, are used to describe various phenomena across mathematics, physics, and engineering. The main hyperbolic functions include the hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), among others. They can be defined in terms of exponential functions:

• \(\text{sinh}(x) = \frac{e^x - e^{-x}}{2}\)
• \(\text{cosh}(x) = \frac{e^x + e^{-x}}{2}\)
• \(\text{tanh}(x) = \frac{\text{sinh}(x)}{\text{cosh}(x)}\)

The beauty of hyperbolic functions lies in their connections to the geometry of a hyperbola, in the same way that trigonometric functions relate to the circle. The exercise provided showcases the use of the hyperbolic cosine function, \(\text{cosh}(x)\), and its importance when dealing with hyperbolic identities.

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are not only limited to the circle-based trigonometric functions but also extend to hyperbolic functions, which share a remarkable resemblance. Identities are used to simplify complex expressions, to solve equations, and to perform integrations and derivations.

Some common trigonometric identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. Mastering these identities, like the hyperbolic cosine identity shown in the exercise, requires practice and understanding of the underlying geometry and algebra. They serve as the building blocks for more advanced topics in mathematics and can often be the key to unlocking a difficult problem.