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Hyperbolic functions are analogs of the familiar trigonometric functions but are based on hyperbolas, rather than circles. They are useful in various areas of mathematics including calculus and complex analysis. Like trigonometric functions, hyperbolic functions have corresponding identities that can simplify equations and solve problems. The most common hyperbolic functions are:

• sinh(x) = \((e^x - e^{-x})/2\), known as the hyperbolic sine function.
• cosh(x) = \((e^x + e^{-x})/2\), known as the hyperbolic cosine function.
• tanh(x) = \(sinh(x) / cosh(x)\), which is \((e^x - e^{-x})/(e^x + e^{-x})\).

Hyperbolic functions provide a means to describe certain geometries and solve differential equations. They share many properties with trigonometric functions, such as similar patterns and identities that relate them to each other.

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities are essential tools in simplifying expressions and solving trigonometric equations. Some of the basic trigonometric identities include:

• Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\).
• Angle sum and difference identities, such as \(\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)\).
• Double angle formulas, like \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

It's important to note that in some exercises, trigonometric identities are combined with hyperbolic identities. This helps in transforming complex equations into more manageable forms. Identifying the right identity can often simplify a solution or reveal hidden relationships within the problem.

Inverse trigonometric functions are the "undoing" of the regular trigonometric functions, allowing you to find the angle that corresponds to a particular trigonometric ratio. They are crucial in solving equations where the variable is an angle but has been transformed using trigonometric functions. The three primary inverse trigonometric functions are:

• \(\arcsin(x)\), which is the inverse of the sine function.
• \(\arccos(x)\), which is the inverse of the cosine function.
• \(\arctan(x)\), which is the inverse of the tangent function.

These functions return the angle, typically in radians, whose trigonometric value equals the given number. For the problem at hand, we use the fact that the expressions involving \(\arctan\) and \(\arcsin\) can be shown to be equal through manipulation of hyperbolic identities. Understanding inverse functions and their properties is key to solving problems that require inverting trigonometric conditions.