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The double angle formula in trigonometry is a pivotal concept, and its hyperbolic counterpart is equally significant in hyperbolic trigonometry. This formula connects the values of hyperbolic functions—hyperbolic cosine (\textbf{cosh}) and hyperbolic sine (\textbf{sinh})—for double the angle. For the hyperbolic cosine, the formula is expressed as: \[\cosh 2x = \cosh^2 x + \sinh^2 x\]
This identity is essential for proving a range of other hyperbolic identities, including the one in our exercise. By understanding the structure and derivation of the double angle formula, learners can more readily tackle problems involving hyperbolic functions.

Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine (\textbf{sinh}) and the hyperbolic cosine (\textbf{cosh}), and they are defined using exponential functions. These definitions are:\[\sinh x = \frac{e^x - e^{-x}}{2}\]\[\cosh x = \frac{e^x + e^{-x}}{2}\]
Hyperbolic functions appear in various areas of mathematics, including in the solutions of certain differential equations and in the descriptions of hyperbolas, hence their name. Just like their circular counterparts, they adhere to similar identity properties which make them useful in simplifying and solving algebraic and calculus problems.

The hyperbolic cosine (\textbf{cosh}) plays a fundamental role in hyperbolic geometry and calculus. It can be visualized on a graph resembling a cosh curve, which is unbounded and symmetric about the y-axis. The function is expressed as:\[\cosh x = \frac{e^x + e^{-x}}{2}\]
This function grows exponentially as x increases or decreases, and it satisfies several identities, including the aforementioned double angle formula. Furthermore, it relates to the hyperbolic sine function by the identity \[\cosh^2 x - \sinh^2 x = 1,\]
which is reminiscent of the Pythagorean identity of trigonometric functions and is critical for transforming and simplifying expressions involving hyperbolic functions.

The hyperbolic sine (\textbf{sinh}) is another core hyperbolic function, showing antisymmetry about the origin and infinite growth as x tends towards positive or negative infinity. Its formula is given by:\[\sinh x = \frac{e^x - e^{-x}}{2}\]
The 'sinh' function, together with 'cosh', helps to describe the shape of a hanging cable or chain, a phenomenon known as a catenary. As with 'cosh', it forms an integral part of various hyperbolic identities and, when manipulated appropriately, allows us to convert expressions involving hyperbolic functions into simpler or more useful forms, as exemplified in our proof of the identity.

Simplifying expressions is a universal technique in mathematics to make them more interpretable or to prepare them for solving. This technique often includes grouping like terms, factoring, expanding, and using known identities to transform expressions into an equivalent form that's easier to work with. In the context of hyperbolic functions, simplifying may involve exploiting relationships between 'sinh' and 'cosh', applying double angle formulas, and recognizing patterns akin to those found in classical trigonometry. By simplifying, we can uncover the underlying structure of complex expressions, making them more manageable and solving equations or proving identities as we've done in this exercise.
Crucially, simplifying can unearth relationships that are not immediately obvious, such as transforming the expression \[\cosh 2x - 1\]
to reveal \[\sinh^2 x.\]