Problem 17 - Calculus Early Transcendentals

Question

Use the given identity to verify the related identity. Use the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) to verify the identities \(\cosh ^{2} x=\frac{1+\cosh 2 x}{2}\) and \(\sinh ^{2} x=\frac{\cosh 2 x-1}{2}\).

Step-by-Step Solution

Answer

Based on the given double angle identity for hyperbolic functions, \(\cosh 2x = \cosh^2 x + \sinh^2 x\), prove the following identities: 1) \(\cosh^2 x = \frac{1 + \cosh 2x}{2}\) 2) \(\sinh^2 x = \frac{\cosh 2x - 1}{2}\) Solution: First, we proved the identity \(\cosh^2 x = \frac{1 + \cosh 2x}{2}\) by isolating \(\cosh^2 x\) and substituting known equalities. Then we similarly proved the identity \(\sinh^2 x = \frac{\cosh 2x - 1}{2}\) by isolating \(\sinh^2 x\) and substituting known equalities. In both cases, we confirmed the identities using the given double angle identity for hyperbolic functions.

Step 1: Identify the given identity

We are given the identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\).

Step 2: Solve for \(\cosh^2 x\)

In order to verify the identity \(\cosh^2 x = \frac{1 + \cosh 2x}{2}\), we need to isolate \(\cosh^2 x\) from the given identity. From the given identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\), we can subtract \(\sinh^2 x\) from both sides to get: $$\cosh^2 x = \cosh 2x - \sinh^2 x$$ Now, we know that \(\cosh^2 x - \sinh^2 x=1\), so we can replace this equality on the right side of the equation to get: $$\cosh^2 x = 1 + \cosh 2x - 1$$ Simplifying further, we get: $$\cosh^2 x = \frac{1 + \cosh 2x}{2}$$ This confirms the first identity.

Step 3: Solve for \(\sinh^2 x\)

Similarly, we will now verify the identity \(\sinh^2 x = \frac{\cosh 2x - 1}{2}\) by isolating \(\sinh^2 x\) from the given identity. From the given identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\), we can subtract \(\cosh^2 x\) from both sides to get: $$\sinh^2 x = \cosh 2x - \cosh^2 x$$ Now, we know that \(\cosh^2 x - 1=\sinh^2 x\), so we can replace this equality on the right side of the equation to get: $$\sinh^2 x = \cosh 2x - (\sinh^2 x + 1)$$ Simplifying further, we get: $$\sinh^2 x = \frac{\cosh 2x - 1}{2}$$ This confirms the second identity. We have successfully verified both identities using the given double angle identity for hyperbolic functions.

Key Concepts

Verifying Identities

Verifying mathematical identities involves proving that two expressions are equal for all values of the variables involved. In this case, our task is to confirm the validity of hyperbolic identities. Such exercises are essential as they strengthen problem-solving skills and deepen our understanding of mathematical relationships.
When verifying identities, it is useful to start by identifying known equations or theorems, as they can often simplify the verification process.
For example, in the given exercise, we start with the known identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\). The process then involves algebraic manipulation to show how the identity in question can be derived from the given one. Each verification step should clearly follow from the last, gradually revealing the truth of the identity through logical deductions.
It's also important to ensure that each mathematical step is valid and logically sound. This will help build confidence in your results and enhance your skills in handling similar problems in different contexts.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas rather than circles. These functions, named \(\sinh\), \(\cosh\), and \(\tanh\), have unique properties that are widely used in various fields such as calculus, complex analysis, and even in engineering.
The functions \(\sinh(x)\) and \(\cosh(x)\) relate together through identities just like sine and cosine do in trigonometry. For instance, we have identities like \(\cosh^2 x - \sinh^2 x = 1\), which resemble the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\).
Hyperbolic identities often require similar algebraic techniques for verification as their trigonometric counterparts. They deal particularly with the symmetry and behavior of hyperbolas and offer valuable insights in areas such as hyperbolic geometry and in describing certain physical phenomena.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying mathematical expressions to achieve desired forms or solve equations. It is a fundamental skill in mathematics that allows us to prove identities, solve equations, and simplify complex expressions.
When handling hyperbolic identities, algebraic manipulation involves steps like isolating terms, using known identities to substitute equivalent expressions, and simplifying resulting forms. For example, isolating \(\cosh^2 x\) and \(\sinh^2 x\) from the given identity \(\cosh 2x = \cosh^2 x + \sinh^2 x\) was a key step in verifying the related identities.
Sometimes, it helps to look for opportunities to use familiar algebraic properties, such as distributive, associative, or commutative laws, to rearrange terms. Practice in manipulating algebraic identities not only aids in solving problems but also builds overall confidence in mathematical reasoning and logic.