Problem 15 - Single Variable Calculus

Question

Prove the identity. \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\)

Step-by-Step Solution

Answer

Using the definitions of hyperbolic sine and cosine functions, we find expressions for \(\sinh(x+y)\) and \(\cosh(x+y)\). Then, we evaluate expressions for the products of \(\cosh x \cdot \cosh y\) and \(\sinh x \cdot \sinh y\) and expand the expressions. By combining the terms and simplifying, we prove the identity \(\cosh (x+y) = \cosh x \cdot \cosh y + \sinh x \cdot \sinh y\).

Step 1: Using the definitions of sinh and cosh, we can find expressions for sinh(x+y) and cosh(x+y): \[\sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2}\] \[\cosh(x+y) = \frac{e^{x+y} + e^{-(x+y)}}{2}\] #Step 2: Simplify sinh(x+y) expression

Using properties of exponents, let's simplify the expression for \(\sinh(x+y)\): \[\sinh(x+y) = \frac{e^x e^y - e^{-x}e^{-y}}{2}\] #Step 3: Simplify cosh(x+y) expression

Step 2: Similarly, let's simplify the expression for \(\cosh(x+y)\): \[\cosh(x+y) = \frac{e^x e^y + e^{-x} e^{-y}}{2}\] #Step 4: Evalute expressions for products of sinh, cosh and their combinations

Now we need to find the expressions for the following combinations using the definitions of sinh and cosh: 1. \(\cosh x \cdot \cosh y\) 2. \(\sinh x \cdot \sinh y\) \[\cosh x \cdot \cosh y = \frac{e^x + e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2}\] \[\sinh x \cdot \sinh y = \frac{e^x - e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2}\] #Step 5: Expand the expressions

Step 3: Now let's expand the expressions in step 4: \[\cosh x \cdot \cosh y = \frac{e^{x+y} + e^{-x+y} + e^{x-y} + e^{-x-y}}{4}\] \[\sinh x \cdot \sinh y = \frac{- e^{-x+y} - e^{x-y} + e^{x+y} + e^{-x-y}}{4}\] #Step 6: Combine terms and prove the identity

Next, let's add the expressions for \(\cosh x \cdot \cosh y\) and \(\sinh x \cdot \sinh y\) and simplify: \[\cosh x \cdot \cosh y + \sinh x \cdot \sinh y = \frac{e^{x+y} + e^{-x+y} + e^{x-y} + e^{-x-y}}{4} + \frac{- e^{-x+y} - e^{x-y} + e^{x+y} + e^{-x-y}}{4}\] \[\cosh x \cdot \cosh y + \sinh x \cdot \sinh y = \frac{2e^{x+y} + 2e^{-x-y}}{4}\] \[\cosh x \cdot \cosh y + \sinh x \cdot \sinh y = \frac{e^{x+y} + e^{-x-y}}{2}\] Now, compare the results of Step 3 and 6: \[\cosh(x+y) = \frac{e^{x+y} + e^{-x-y}}{2}\] \[\cosh x \cdot \cosh y + \sinh x \cdot \sinh y = \frac{e^{x+y} + e^{-x-y}}{2}\] As both sides are equal, we've proven the identity: \[\cosh (x+y) = \cosh x \cdot \cosh y + \sinh x \cdot \sinh y\]

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the well-known trigonometric functions but based on hyperbolas instead of circles. These functions have significant applications in calculus, complex numbers, and various branches of physics and engineering.
They often appear when solving linear differential equations and in calculations involving exponential functions.

• Sinh, or hyperbolic sine, which we'll discuss more in depth, is the hyperbolic counterpart of the sine function.
• Cosh, or hyperbolic cosine, resembles the cosine function but has its unique properties and uses.

The hyperbolic functions are defined using exponential functions, which is why we see a considerable resemblance in their structure with the familiar trigonometric identities.
Understanding these connections allows us to apply them in solving various mathematical problems effectively.

Sinh

The hyperbolic sine function, denoted as sinh, is defined in terms of exponential functions.
Its definition is \[\sinh(x) = \frac{e^x - e^{-x}}{2}\] This definition makes it clear that sinh is closely tied to exponential growth and decay.

• The simple structure enables us to express combinations like sinh(x+y) using properties of exponents, leading to higher-level simplifications.
• One notable identity involves the addition formula, which can be proven using the exponential formula: \[\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y\]

The \'hyperbolic sine\' function's unique growth pattern means it increases more quickly than the regular sine, especially for larger values of x.
This behavior is crucial in contexts where modeling rapid changes or reactions in systems is needed.

Cosh

The hyperbolic cosine function, abbreviated as cosh, is another critical hyperbolic function.
Its defining formula via exponential expressions is:\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]This definition highlights its symmetry and stability.

• Cosh is even, meaning \(\cosh(-x) = \cosh(x)\), which provides unique symmetry around the y-axis.
• It is frequently used in expressing hyperbolic identities and modeling phenomena like hanging cables (catenary) and certain mechanical systems.

Like sinh, cosh has addition formulas that involve both sinh and cosh, forming identities used frequently in calculus and physics.
Understanding cosh's behavior is essential, given its role in shaping various physical and theoretical models in science and engineering.

Exponential Functions

Exponential functions are foundational for understanding hyperbolic functions.
These functions are expressed in the form \(e^x\), where \(e\) is Euler's number, approximately 2.718.

• Exponential functions describe growth and decay processes, from population increases to radioactive decay.
• Their properties, like \(e^{a+b} = e^a \cdot e^b\), are pivotal in simplifying expressions in calculus and hyperbolic function proofs.

With hyperbolic functions, exponential functions help define their shapes and behaviors, as seen from their defining formulas.
When you see identities involving hyperbolic functions, they are often reduced to manipulating exponential terms, making their mastery crucial for deeper mathematical studies.