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The hyperbolic tangent function, denoted as \( \tanh(z) \), is a key hyperbolic function used in many areas of mathematics and engineering. It's similar to the tangent function from trigonometry but is derived from hyperbolic sine (\( \sinh \)) and hyperbolic cosine (\( \cosh \)) functions. For any variable \( z \), \( \tanh(z) = \frac{\sinh(z)}{\cosh(z)} \). This means the hyperbolic tangent is the ratio of the hyperbolic sine to the hyperbolic cosine of \( z \). It's a function that smoothly varies between -1 and 1, unlike its trigonometric counterpart, which oscillates between negative and positive infinity. This makes \( \tanh \) a very important function in fields such as calculus, where it is appreciated for its smooth nature and its asymptotic behavior close to 1 and -1 as \( z \) tends to infinity.In the context of the original exercise, proving the identity \( \tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \) involves leveraging the properties of \( \tanh \). This identity bears resemblance to the angle addition formula in trigonometry but applied to hyperbolic functions.

Hyperbolic identities are equations that relate hyperbolic functions to one another. Similar to trigonometric identities, they offer ways to simplify and transform expressions. For the hyperbolic tangent function, such identities can be particularly useful when dealing with sums or differences of arguments.The original exercise utilizes a key hyperbolic identity for \( \tanh(x+y) \), similar to the addition formulas in trigonometry. The identity \( \tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \) is used in the solution to transform a complex expression into a more manageable form. This identity shows how the function of a sum can be expressed in terms of the individual functions, also illustrating the relationship between compound arguments and individual arguments in hyperbolic context.Hyperbolic identities are indispensable tools in calculus and algebra, particularly for integration and simplification of expressions involving hyperbolic functions. Mastering these can ease solving of complex mathematical problems, particularly in the fields of engineering and physics.

Hyperbolic sine and cosine, abbreviated as \( \sinh \) and \( \cosh \) respectively, are basic hyperbolic functions. They are defined as follows:

• \( \sinh(z) = \frac{e^z - e^{-z}}{2} \)
• \( \cosh(z) = \frac{e^z + e^{-z}}{2} \)

These definitions are analogous to their trigonometric counterparts but are based on exponential functions rather than circular functions.\( \sinh \) and \( \cosh \) are building blocks for expressing the hyperbolic tangent function, as well as providing key formulas in the solution to the exercise. When proving the identity given in the exercise, the use of the sum formulas for \( \sinh(x+y) \) and \( \cosh(x+y) \) enables the transformation of the expression into a form that can be easily simplified.These sum formulas are as follows:

• \( \sinh(x+y) = \sinh(x)\cosh(y) + \cosh(x)\sinh(y) \)
• \( \cosh(x+y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y) \)

By applying these formulas, one can derive more complex hyperbolic identities and solve hyperbolic equations efficiently. These functions regularly appear in problems involving hyperbolic geometry, calculus, and complex analysis.