Problem 11 - Thomas Calculus

Question

Use the identities $$ \begin{aligned} \sinh (x+y) &=\sinh x \cosh y+\cosh x \sinh y \\ \cosh (x+y) &=\cosh x \cosh y+\sinh x \sinh y \end{aligned} $$ to show that a. \(\sinh 2 x=2 \sinh x \cosh x\) b. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

Step-by-Step Solution

Answer

a. \(sinh(2x) = 2sinh(x)\cosh(x)\). b. \(cosh(2x) = cosh^2(x) + sinh^2(x)\).

Step 1: Use the Identity for Sinh

To show the result for part (a), use the identity for hyperbolic sine: \[sinh(x+y) = sinh(x)\cosh(y) + cosh(x)\sinh(y) \]Set \( y = x \) to find \( sinh(2x) \). The identity becomes:\[sinh(2x) = sinh(x)\cosh(x) + cosh(x)\sinh(x) \]

Step 2: Simplify the Expression for Sinh

Combine the terms from the identity:\[sinh(2x) = sinh(x)\cosh(x) + cosh(x)\sinh(x) = 2sinh(x)\cosh(x) \]This shows the desired expression for \( sinh(2x) \).

Step 3: Use the Identity for Cosh

To show the result for part (b), use the identity for hyperbolic cosine:\[cosh(x+y) = cosh(x)\cosh(y) + sinh(x)\sinh(y) \]Set \( y = x \) to find \( cosh(2x) \). The identity becomes:\[cosh(2x) = cosh(x)\cosh(x) + sinh(x)\sinh(x) \]

Step 4: Simplify the Expression for Cosh

Combine the terms from the identity:\[cosh(2x) = cosh^2(x) + sinh^2(x) \]This shows the desired expression for \( cosh(2x) \).

Key Concepts

Hyperbolic Identities

Hyperbolic identities are essential tools when dealing with hyperbolic functions. They are analogous to trigonometric identities.
These identities help in simplifying complex expressions involving hyperbolic sine (\sinh) and hyperbolic cosine (\cosh) functions. Two vital hyperbolic identities given are:

• \( \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \)
• \( \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \)

These identities allow us to understand the behavior of hyperbolic functions better, especially how they interact when added together.
By using these identities, complex hyperbolic expressions can be rewritten in simpler forms, just like the step-by-step solution for the problems showed for \(\sinh 2x\) and \(\cosh 2x\). This simplification is essentially achieved through substitution and basic algebraic manipulation.

Sinh Function

The \(\sinh\) function, short for hyperbolic sine, is an analogy to the trigonometric sine function but within the realm of hyperbolas.
The definition of the \(\sinh\) function is given by:

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

This function takes input on the real number line and outputs a value that can be thought of as a measure related to a hyperbola.
In the specific exercise, using the identity \(\sinh(x+y)\), when we set \(y = x\), the identity becomes \(\sinh 2x = 2 \sinh x \cosh x\).
This shows how the \(\sinh\) function behaves when its input is doubled. By recognizing the repetition of the term \(\sinh x \cosh x\), the solution becomes elegant and clear.

Cosh Function

The \( \cosh \) function, known as hyperbolic cosine, parallels the trigonometric cosine function but relates to hyperbolas.
Defined as:

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

The \(\cosh\) function represents a symmetrical variant about the y-axis.
In the problem context, setting \( y = x \) in the identity \( \cosh(x + y) \), we discover that \( \cosh 2x = \cosh^2 x + \sinh^2 x \).
This formulation neatly illustrates the inherent properties of the \( \cosh \) function and how it ties with its counterpart, the \( \sinh \) function. This identity also shows us a peek into why hyperbolic functions appear similar yet distinctly different from their trigonometric counterparts.