Problem 29 - Advanced Engineering Mathematics

Question

$$ \text { Prove that } \cosh ^{2} z-\sinh ^{2} z=1 \text {. } $$

Step-by-Step Solution

Answer

The identity \(\cosh^2 z - \sinh^2 z = 1\) holds true, as proven with hyperbolic function definitions.

Step 1: Recall Definitions

Recall the definitions of hyperbolic functions: \[\cosh z = \frac{e^z + e^{-z}}{2} \] and \[\sinh z = \frac{e^z - e^{-z}}{2}.\] These definitions will be important for proving the identity.

Step 2: Apply Cosine Hyperbolic to the Left Side

Calculate \(\cosh^2 z\) by squaring the definition of hyperbolic cosine: \[\cosh^2 z = \left(\frac{e^z + e^{-z}}{2}\right)^2.\] Expanding the square: \[\cosh^2 z = \frac{e^{2z} + 2 + e^{-2z}}{4}.\]

Step 3: Apply Sine Hyperbolic to the Left Side

Calculate \(\sinh^2 z\) by squaring the definition of hyperbolic sine: \[\sinh^2 z = \left(\frac{e^z - e^{-z}}{2}\right)^2.\] Expanding the square: \[\sinh^2 z = \frac{e^{2z} - 2 + e^{-2z}}{4}.\]

Step 4: Substitute and Simplify

Substitute \(\cosh^2 z\) and \(\sinh^2 z\) into the given identity \(\cosh^2 z - \sinh^2 z = 1\):\[\frac{e^{2z} + 2 + e^{-2z}}{4} - \frac{e^{2z} - 2 + e^{-2z}}{4}.\]Simplify the expression by combining terms: \[= \frac{(e^{2z} + 2 + e^{-2z}) - (e^{2z} - 2 + e^{-2z})}{4} = \frac{4}{4} = 1.\]

Step 5: Conclusion

Thus, the identity \(\cosh^2 z - \sinh^2 z = 1\) is proven by substituting definitions and simplifying the expressions.

Key Concepts

Hyperbolic Identities

Hyperbolic identities are analogous to trigonometric identities, but they relate to hyperbolic functions instead of trigonometric ones. One of the basic hyperbolic identities is \( \cosh^2 z - \sinh^2 z = 1 \). This identity is fundamental because it resembles the Pythagorean identity in trigonometry, where \( \cos^2 \theta + \sin^2 \theta = 1 \).
However, unlike trigonometric functions, hyperbolic functions are defined using exponential functions and are relevant in hyperbolic geometry and complex analysis.

Here’s how to prove this identity:

• Recall the definitions \( \cosh z = \frac{e^z + e^{-z}}{2} \) and \( \sinh z = \frac{e^z - e^{-z}}{2} \).
• Square both expressions to find \( \cosh^2 z \) and \( \sinh^2 z \).
• Subtract the squared values to establish the hyperbolic identity.

What is intriguing about these identities is how they link back to important applications in mathematics, transcending simple exponential functions to explain phenomena like the shape of a hanging cable (catenary) or relativistic motion in physics.

Hyperbolic Cosine

The hyperbolic cosine function, denoted as \( \cosh z \), is defined by the formula \( \cosh z = \frac{e^z + e^{-z}}{2} \). This function describes the shape of an idealized hanging cable or chain, which is why it is often associated with a catenary curve.
Its properties are similar to those of the usual cosine function but are distinguished by its definition in terms of exponential functions.

Some key features of the hyperbolic cosine include:

• Even Function: \( \cosh(-z) = \cosh(z) \), meaning it is symmetric about the \( y \)-axis.
• Always Positive: \( \cosh(z) > 0 \) for all real numbers \( z \).
• Approaches infinity as \( z \) approaches infinity in both directions.

The behavior of \( \cosh z \) makes it significant in mathematical fields like calculus for solving differential equations and in physics for describing energy potentials and wave equations.

Hyperbolic Sine

The hyperbolic sine function, denoted as \( \sinh z \), is defined as \( \sinh z = \frac{e^z - e^{-z}}{2} \). This function, much like its trigonometric counterpart, illustrates a relationship with hyperbolas that is equivalent to that of the sine function with circles.
The graphs of \( \sinh z \) portray an increasing nature, similar to the sine but unbounded and more linear.

Key properties of hyperbolic sine are:

• Odd Function: \( \sinh(-z) = -\sinh(z) \), indicating symmetry about the origin.
• Range: Values can be both positive and negative, and \( \sinh(z) \) ranges from \(-\infty\) to \(\infty\).
• The slope at the origin is 1, and it increases exponentially for large \( z \).

These properties make \( \sinh z \) not only mathematically fascinating but also useful in physics, especially in scenarios involving exponential growth processes and in the hyperbolic descriptions of spatial regions.