Chapter 2: Problem 47
Verify that the given equations are identities. \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\)
Short Answer
Expert verified
The equation is verified as an identity.
Step by step solution
01
Understand Hyperbolic Identities
Recall that hyperbolic identities are similar to trigonometric identities, but they use hyperbolic functions. Hyperbolic functions include \(\sinh\) and \(\cosh\). The specific identity we are verifying is \(\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y\).
02
Use Definitions of Hyperbolic Functions
The hyperbolic cosine is defined as \(\cosh x = \frac{e^x + e^{-x}}{2}\) and hyperbolic sine as \(\sinh x = \frac{e^x - e^{-x}}{2}\). Use these definitions to express \(\cosh(x+y)\), \(\cosh x \cosh y\), and \(\sinh x \sinh y\) in terms of exponential functions.
03
Expand \(\cosh(x+y)\) Using Exponential Functions
Expand \(\cosh(x+y)\) using its definition: \[\cosh(x+y) = \frac{e^{x+y} + e^{-(x+y)}}{2} = \frac{e^x e^y + e^{-x} e^{-y}}{2}.\]
04
Expand \(\cosh x \cosh y\) and \(\sinh x \sinh y\)
Calculate \(\cosh x \cosh y\) using definitions:\[\cosh x \cosh y = \left(\frac{e^x + e^{-x}}{2}\right)\left(\frac{e^y + e^{-y}}{2}\right) = \frac{e^{x+y} + e^{x-y} + e^{-x+y} + e^{-(x+y)}}{4}\]Calculate \(\sinh x \sinh y\):\[\sinh x \sinh y = \left(\frac{e^x - e^{-x}}{2}\right)\left(\frac{e^y - e^{-y}}{2}\right) = \frac{e^{x+y} - e^{x-y} - e^{-x+y} + e^{-(x+y)}}{4}\]
05
Combine Results
Add the expansions from step 4:\[\cosh x \cosh y + \sinh x \sinh y = \frac{e^{x+y} + e^{x-y} + e^{-x+y} + e^{-(x+y)}}{4} + \frac{e^{x+y} - e^{x-y} - e^{-x+y} + e^{-(x+y)}}{4} = \frac{2e^{x+y} + 2e^{-(x+y)}}{4} = \frac{e^{x+y} + e^{-(x+y)}}{2}\]This simplifies exactly to \(\cosh(x+y)\) from step 3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbolic Identities
Hyperbolic identities play a key role in simplifying expressions related to hyperbolic functions. Just like trigonometric identities simplify trigonometric expressions, hyperbolic identities do the same for hyperbolic functions, such as \(\sinh\) and \(\cosh\). These identities often appear in algebraic and calculus problems, presenting an alternative to familiar trigonometric identities. For instance, the identity \(\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y\) resembles the addition formulas from trigonometry but applies to hyperbolic functions.
These identities can transform complex expressions into more manageable forms, facilitating easier analysis and solution of mathematical problems. Understanding these identities allows students to seamlessly transition between trigonometric and hyperbolic contexts, offering deeper insights and a broader toolkit for tackling mathematical challenges.
These identities can transform complex expressions into more manageable forms, facilitating easier analysis and solution of mathematical problems. Understanding these identities allows students to seamlessly transition between trigonometric and hyperbolic contexts, offering deeper insights and a broader toolkit for tackling mathematical challenges.
Cosh and Sinh Definitions
The hyperbolic cosine and sine functions, \(\cosh\) and \(\sinh\), respectively, are fundamental to understanding hyperbolic identities.
These functions are defined in terms of exponential functions, offering a beautiful intersection between algebra and calculus.
- The hyperbolic cosine is defined as:
\[ \cosh x = \frac{e^x + e^{-x}}{2} \]
- The hyperbolic sine is defined as:
\[ \sinh x = \frac{e^x - e^{-x}}{2} \]
These definitions are crucial, as they not only help in manipulating hyperbolic expressions but also in integrating and differentiating hyperbolic functions. By expressing \(\cosh\) and \(\sinh\) in terms of exponentials, one can leverage exponential function properties, leading to potential simplifications or transformations that might not be immediately apparent using only the hyperbolic function notation. Thus, grasping these definitions opens numerous doors in mathematical problem-solving and allows for more powerful techniques in calculus and algebra.
These functions are defined in terms of exponential functions, offering a beautiful intersection between algebra and calculus.
- The hyperbolic cosine is defined as:
\[ \cosh x = \frac{e^x + e^{-x}}{2} \]
- The hyperbolic sine is defined as:
\[ \sinh x = \frac{e^x - e^{-x}}{2} \]
These definitions are crucial, as they not only help in manipulating hyperbolic expressions but also in integrating and differentiating hyperbolic functions. By expressing \(\cosh\) and \(\sinh\) in terms of exponentials, one can leverage exponential function properties, leading to potential simplifications or transformations that might not be immediately apparent using only the hyperbolic function notation. Thus, grasping these definitions opens numerous doors in mathematical problem-solving and allows for more powerful techniques in calculus and algebra.
Exponential Functions in Hyperbolic Functions
Exponential functions are foundational in defining hyperbolic functions and in proving hyperbolic identities.
By using exponential expressions, you can break down hyperbolic functions into more understandable components. Both \(\cosh(x+y)\) and expressions like \(\cosh x \cosh y\) or \(\sinh x \sinh y\) can be broken down using their exponential definitions.
Here's how each is expressed with exponentials: - For \(\cosh(x+y)\):
\[ \cosh(x+y) = \frac{e^{x+y} + e^{-(x+y)}}{2} = \frac{e^x e^y + e^{-x} e^{-y}}{2} \]- For \(\cosh x \cosh y\):
\[ \cosh x \cosh y = \frac{e^{x+y} + e^{x-y} + e^{-x+y} + e^{-(x+y)}}{4} \]- For \(\sinh x \sinh y\):
\[ \sinh x \sinh y = \frac{e^{x+y} - e^{x-y} - e^{-x+y} + e^{-(x+y)}}{4} \]
These expressions showcase that hyperbolic identities often boil down to exponential identities, highlighting the deep connection between different areas of mathematics. This intersection provides a seamless understanding of identities and enhances the flexibility in manipulating and simplifying expressions or proving the equivalence between different mathematical statements.
By using exponential expressions, you can break down hyperbolic functions into more understandable components. Both \(\cosh(x+y)\) and expressions like \(\cosh x \cosh y\) or \(\sinh x \sinh y\) can be broken down using their exponential definitions.
Here's how each is expressed with exponentials: - For \(\cosh(x+y)\):
\[ \cosh(x+y) = \frac{e^{x+y} + e^{-(x+y)}}{2} = \frac{e^x e^y + e^{-x} e^{-y}}{2} \]- For \(\cosh x \cosh y\):
\[ \cosh x \cosh y = \frac{e^{x+y} + e^{x-y} + e^{-x+y} + e^{-(x+y)}}{4} \]- For \(\sinh x \sinh y\):
\[ \sinh x \sinh y = \frac{e^{x+y} - e^{x-y} - e^{-x+y} + e^{-(x+y)}}{4} \]
These expressions showcase that hyperbolic identities often boil down to exponential identities, highlighting the deep connection between different areas of mathematics. This intersection provides a seamless understanding of identities and enhances the flexibility in manipulating and simplifying expressions or proving the equivalence between different mathematical statements.
Verifying Mathematical Identities
Verifying mathematical identities, especially in hyperbolic functions, is a critical skill that enables one to confirm the validity of what might initially appear as complex equations.
In the given exercise, verifying the identity \(\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y\) involves expanding both sides using their exponential definitions and simplifications. - First, expand and simplify \(\cosh(x+y)\)- Next, perform similar expansions and simplifications for \(\cosh x \cosh y\) and \(\sinh x \sinh y\).- Finally, add these expressions to check if they simplify to the expression obtained for \(\cosh(x+y)\).
When both sides of a given equation simplify to the same exact form, as shown with exponential manipulations, it confirms that the identity holds true. This process reinforces the understanding of both exponential and hyperbolic functions, fostering the ability to prove or derive formulas independently. Successfully verifying identities underlines proficiency in mathematical reasoning and builds confidence in handling complex algebraic manipulations.
In the given exercise, verifying the identity \(\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y\) involves expanding both sides using their exponential definitions and simplifications. - First, expand and simplify \(\cosh(x+y)\)- Next, perform similar expansions and simplifications for \(\cosh x \cosh y\) and \(\sinh x \sinh y\).- Finally, add these expressions to check if they simplify to the expression obtained for \(\cosh(x+y)\).
When both sides of a given equation simplify to the same exact form, as shown with exponential manipulations, it confirms that the identity holds true. This process reinforces the understanding of both exponential and hyperbolic functions, fostering the ability to prove or derive formulas independently. Successfully verifying identities underlines proficiency in mathematical reasoning and builds confidence in handling complex algebraic manipulations.
