Problem 12 - Single Variable Calculus

Question

Prove the identity. \(\sinh 2 x=2 \sinh x \cosh x\)

Step-by-Step Solution

Answer

Using the definitions of hyperbolic sine and cosine functions, we compute the expressions for \(\sinh(2x)\) and \(2 \sinh x \cosh x\). Upon simplification, we find that both expressions are equal, thus proving the identity: \(\sinh 2x = 2 \sinh x \cosh x\).

Step 1: Recall the definitions of \(\sinh\) and \(\cosh\)

The hyperbolic sine and cosine functions are defined as follows: \(\sinh x = \dfrac{e^x - e^{-x}}{2}\) \(\cosh x = \dfrac{e^x + e^{-x}}{2}\) Note that these definitions are derived from the exponential functions, and are used to express the given identity in terms of exponential functions.

Step 2: Compute the expression for \(\sinh(2x)\)

Using the definition of the hyperbolic sine function, we can compute the expression for \(\sinh(2x)\) as follows: \(\sinh(2x) = \dfrac{e^{2x} - e^{-2x}}{2}\)

Step 3: Compute the expression for \(2 \sinh x \cosh x\)

Using the definitions of the hyperbolic sine and cosine functions, we can compute the expression for \(2 \sinh x \cosh x\) as follows: \(2 \sinh x \cosh x = 2 \left(\dfrac{e^x - e^{-x}}{2}\right) \left(\dfrac{e^x + e^{-x}}{2}\right)\) Now, let's simplify the expression: \(2 \sinh x \cosh x = (e^x - e^{-x})(e^x + e^{-x})\)

Step 4: Show that both expressions are equal

Now, we will show that both expressions we derived in Step 2 and Step 3 are equal: \(\sinh(2x) = \dfrac{e^{2x} - e^{-2x}}{2}\) \(2 \sinh x \cosh x = (e^x - e^{-x})(e^x + e^{-x})\) Let's expand the expression for \(2 \sinh x \cosh x\): \(2 \sinh x \cosh x = e^{2x} + e^{-2x} - e^{2x} - e^{-2x}\) Now, we can see that both expressions are equal: \(\sinh(2x) = 2 \sinh x \cosh x\) Hence, we have proved the identity: \(\sinh 2x = 2 \sinh x \cosh x\)

Key Concepts

Identity Proof

Proving mathematical identities can seem tricky, but they help reinforce the use of definitions and algebraic manipulation. To prove an identity, we need to show that both sides of an equation represent the same numerical value for all values of the variable involved. In the case of hyperbolic functions like \(\sinh\) and \(\cosh\), we use their exponential definitions to guide our proof.
The process typically involves

• Expressing each function using its definition.
• Simplifying or transforming each side of the equation to reveal their equality.

For instance, to prove the identity \(\sinh(2x) = 2\sinh(x)\cosh(x)\), we start by expressing \(\sinh(2x)\) using the exponential definitions. Then, transform the right side of the equation using those same definitions.

Through careful expansion and simplification, we'll see that the identity holds true for any value of \(x\). It's like solving a puzzle, where our job is to make every piece fit perfectly to complete the equation.

Hyperbolic Sine

The hyperbolic sine function, represented as \(\sinh(x)\), is analogous to the sine function in trigonometry but built with exponential functions. It is defined as:
\[ \sinh x = \dfrac{e^x - e^{-x}}{2} \]
Key characteristics of \(\sinh\) include:

• It represents the imaginary part of a complex number when expressed in terms of \(e^x\).
• Unlike the trigonometric sine, \(\sinh(x)\) can take on any real number value.
• \sinh(x) has no limit as \(x\) becomes very large or very small.

The function is used in various fields, including physics and engineering, especially when analyzing hyperbolic geometry and problems involving hyperbolic equations. To better understand it, you can plot the graph of \(\sinh(x)\). You’ll notice its graph is symmetric about the origin, reflecting its odd function nature.

Hyperbolic Cosine

The hyperbolic cosine function, shown as \(\cosh(x)\), shares conceptual similarities with the trigonometric cosine function, but like \(\sinh(x)\), it is rooted in exponentials. Its definition is:
\[ \cosh x = \dfrac{e^x + e^{-x}}{2} \]
Useful attributes of \(\cosh\) include:

• It calculates the average of two exponential functions, providing stability.
• Unlike \(\sinh(x)\), \(\cosh(x)\) embodies even function properties, meaning the graph is symmetric about the y-axis.
• The smallest value of \(\cosh(x)\) is 1, which occurs at \(x = 0\).

The function is crucial in physics, often appearing in the study of equations such as the wave equation and general relativity. Similar to \(\sinh(x)\), you can plot \(\cosh(x)\) to see its characteristic U-shape. Understanding how \(\sinh\) and \(\cosh\) interrelate through identities like \(\sinh^2(x) - \cosh^2(x) = -1\) further unlocks the geometry of complex analysis.