Question

Prove the hyperbolic identity $$ \cosh 2 x=\sinh ^{2} x+\cosh ^{2} x $$

Short answer

Question: Prove the hyperbolic identity: \(\cosh 2x = \sinh^2x + \cosh^2x\). Answer: Following the steps from the above solution, we rewrite the given identity using the definitions of hyperbolic sine and cosine functions and then simplify the expressions. After comparing the expressions, we find that \(\cosh 2x = \sinh^2x + \cosh^2x\), thus proving the hyperbolic identity.

Step-by-step solution

Write down the given identity

The given identity is: $$ \cosh 2x = \sinh^2 x + \cosh^2 x $$

Recall the definitions of hyperbolic functions

The hyperbolic sine and cosine functions are defined as: $$ \sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2} $$

Apply the definition of hyperbolic cosine to 2x

We apply the definition of hyperbolic cosine to \(\cosh 2x\): $$ \cosh 2x = \frac{e^{2x} + e^{-2x}}{2} $$

Write the squares of hyperbolic sine and cosine functions

We write the squares of \(\sinh x\) and \(\cosh x\) and use their definitions: $$ \sinh^2x = \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{e^{2x} - 2 + e^{-2x}}{4} $$ $$ \cosh^2x = \left(\frac{e^x + e^{-x}}{2}\right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4} $$

Add the squares of hyperbolic sine and cosine

Add the expressions for \(\sinh^2x\) and \(\cosh^2x\): $$ \sinh^2x + \cosh^2x = \frac{e^{2x} - 2 + e^{-2x}}{4} + \frac{e^{2x} + 2 + e^{-2x}}{4} = \frac{2e^{2x}+2e^{-2x}}{4} = \frac{e^{2x} + e^{-2x}}{2} $$

Compare the expressions

Recall that we found \(\cosh 2x=\frac{e^{2x}+e^{-2x}}{2}\) in step 3 and \(\sinh^2x+\cosh^2x=\frac{e^{2x}+e^{-2x}}{2}\) in step 5. Since both expressions are equal, the given hyperbolic identity is proven: $$ \cosh 2x = \sinh^2x + \cosh^2x $$

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions. However, instead of being related to circles, they are linked to hyperbolas. These functions typically appear in the context of calculus, physics, and engineering. The primary hyperbolic functions are hyperbolic sine (\(\sinh x\)) and hyperbolic cosine (\(\cosh x\)). Here are their definitions:

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

These functions have unique properties and identities, much like their trigonometric counterparts. They play a crucial role in modeling phenomena such as catenaries and natural exponential growth. Understanding how these functions work and how to manipulate them is essential for many advanced mathematical applications. Often, knowing the standard identities and properties helps you solve more complex problems involving hyperbolic functions. The identity in our exercise is just one example of how hyperbolic functions can be analyzed.

Mathematical Proofs

Mathematical proofs act as a backbone for verifying mathematical statements. In this exercise, we used a proof to verify the hyperbolic identity\(\cosh 2x = \sinh^2 x + \cosh^2 x\). Proofs begin by using definitions and known identities to logically demonstrate the truth of a new statement.
To prove the hyperbolic identity, we:

• Started with the definitions of hyperbolic functions.
• Calculated the expressions for \(\cosh 2x\) and the sum \(\sinh^2x + \cosh^2x\).
• Shown that both expressions equate to the same result.

In mathematics, especially during undergraduate studies, crafting proofs deepens understanding and trains logical thinking skills. Proofs require attention to detail and a step-by-step approach to build a compelling argument for verifying or disproving mathematical claims.

Undergraduate Mathematics

Undergraduate mathematics covers a wide range of topics, including calculus, algebra, geometry, and discrete mathematics. It lays the foundation for more advanced mathematical ideas and technical fields like engineering, computer science, and economics.
At this level, students are introduced to new function types such as hyperbolic functions and are encouraged to understand their behaviors and applications. Learning to prove mathematical identities, as shown in our exercise, helps cement this understanding. These skills are not only crucial for passing exams but are applicable in practical scenarios like coding simulations and solving physical equations.
Navigating through undergraduate mathematics involves mastering the basics while also being adaptable to tackle less straightforward topics. It is a stimulating phase in mathematical education that pushes students to develop abstract thinking and problem-solving abilities.