Question

Show that (1) \(\cos h^{2} x-\sin h^{2} x=1\) (2) \(\sin \mathrm{h}(\mathrm{x}+\mathrm{y})=\sin \mathrm{h} \mathrm{x} \cos \mathrm{h} \mathrm{y}+\cos \mathrm{h} \mathrm{x} \sin \mathrm{h} \mathrm{y}\)

Short answer

We can prove the given identities by using the definitions of hyperbolic functions and their properties: (1) The first identity, \(\cosh^2{x} - \sinh^2{x} = 1\), is proven by substituting the definitions of \(\sinh{x} = \frac{e^x - e^{-x}}{2}\) and \(\cosh{x} = \frac{e^x + e^{-x}}{2}\) and simplifying the expression, which results in \(\frac{4}{4} = 1\). (2) The second identity, \(\sinh{(x + y)} = \sinh{x} \cosh{y} + \cosh{x} \sinh{y}\), is proven by applying the addition formula for hyperbolic functions and substituting the definitions of \(\sinh{x}\) and \(\cosh{x}\) as mentioned in (1). Upon simplifying the expression, we obtain the same left-hand side and right-hand side, verifying the identity.

Step-by-step solution

Recall definitions of hyperbolic functions

The hyperbolic sine and cosine functions are defined as follows: \[\sinh{x} = \frac{e^x - e^{-x}}{2}\] \[\cosh{x} = \frac{e^x + e^{-x}}{2}\]

Prove the first identity

We will substitute the definitions of \(\sinh{x}\) and \(\cosh{x}\) into the given equation: \[\cosh^2{x} - \sinh^2{x} = 1\] Using the definitions from step 1, we get: \[\left(\frac{e^x + e^{-x}}{2}\right)^2 - \left(\frac{e^x - e^{-x}}{2}\right)^2 = 1\] Now let's simplify this expression by expanding the squares and combining the terms. \[\frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4} = 1\] Subtract the two fractions: \[\frac{e^{2x} + 2 + e^{-2x} - e^{2x} + 2 - e^{-2x}}{4} = 1\] Cancelling out the terms \(e^{2x}\) and \(e^{-2x}\) leaves: \[\frac{4}{4} = 1\] Which is true, so the first identity \(\cosh^2{x} - \sinh^2{x} = 1\) is proven.

Prove the second identity

To prove this identity, we will use the addition formula for hyperbolic functions, which is as follows: \[\sinh(x + y) = \sinh{x} \cosh{y} + \cosh{x} \sinh{y}\] Now we can substitute the definitions of \(\sinh{x}\) and \(\cosh{x}\) from step 1, into this formula, to get: \[\frac{e^{x + y} - e^{-(x + y)}}{2} = \frac{e^x - e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} + \frac{e^x + e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2}\] We can simplify this expression by multiplying the fractions on the right-hand side: \[\frac{e^{x + y} - e^{-(x + y)}}{2} = \frac{(e^x - e^{-x})(e^y + e^{-y}) + (e^x + e^{-x})(e^y - e^{-y})}{4}\] Now, expand the terms on the right-hand side: \[\frac{e^x e^y + e^x e^{-y} - e^{-x} e^y - e^{-x} e^{-y}}{4} + \frac{e^x e^y - e^x e^{-y} + e^{-x} e^y - e^{-x} e^{-y}}{4}\] Combine the two fractions and notice that the terms in the middle cancel each other out: \[\frac{2e^x e^y - 2e^{-x} e^{-y}}{4}\] Which is equivalent to: \[\frac{e^{x + y} - e^{-(x + y)}}{2}\] Thus, the second identity \(\sinh{(x + y)} = \sinh{x} \cosh{y} + \cosh{x} \sinh{y}\) is proven.

Key concepts

Hyperbolic Identities

Hyperbolic identities are very similar to trigonometric identities that you may already know, such as \(\cos^2{x} + \sin^2{x} = 1\). Instead, the hyperbolic identity expressing a key relationship is \(\cosh^2{x} - \sinh^2{x} = 1\). This identity is foundational in hyperbolic functions, often referred to as hyperbolic Pythagorean identity, analogous to its trigonometric counterpart.

Understanding this identity can be immensely useful for simplifying expressions involving hyperbolic functions. You can use this identity much like you use trigonometric identities to solve various problems, including integration, differentiation, and simplification of complex hyperbolic expressions. By applying this identity, transformations between different hyperbolic functions become more straightforward, and they form an essential part of advanced mathematics and physics.

sinh

The hyperbolic sine, or \(\sinh\), is one of the basic building blocks of hyperbolic functions. Its definition is given by the formula \(\sinh{x} = \frac{e^x - e^{-x}}{2}\). This formula highlights the exponential nature of hyperbolic functions, which is quite different from the sinusoidal features of regular sine and cosine.

What does \(\sinh{x}\) represent? It can be thought of as the vertical coordinate of a point on the unit hyperbola, much like the regular sine function represents vertical distance on a unit circle.

• The function is odd, meaning \(\sinh{-x} = -\sinh{x}\).
• It's also continuously increasing, demonstrating that it captures exponential growth and decay.

Understanding the properties of \(\sinh\) allows you to use it effectively in solving equations and modeling hyperbolic phenomena, such as hanging cables (catenary) and the behavior of electric currents.

cosh

The hyperbolic cosine, known as \(\cosh\), pairs with \(\sinh\) as fundamental hyperbolic functions. It's defined by the expression \(\cosh{x} = \frac{e^x + e^{-x}}{2}\). This definition illustrates that \(\cosh{x}\) always yields positive values - a feature that makes it similar yet distinct from its trigonometric cousin.

Unlike the trigonometric cosine, \(\cosh\) is an even function, as it satisfies \(\cosh{-x} = \cosh{x}\). This symmetry about the y-axis enables specific transformations and calculations to remain straightforward, often used in physics and engineering.

• \(\cosh{x}\) approaches the exponential function \(e^x\) as \(x\) becomes large.
• It defines the shape of a catenary curve, the curve of a hanging flexible cable.

The understanding of \(\cosh\) is vital for problems involving exponential growth and decay, particularly when dealing with hyperbolic angles and surfaces.

Addition Formula for Hyperbolic Functions

The addition formula for hyperbolic functions provides a way to simplify expressions where sums of arguments appear. For the hyperbolic sine, it is expressed as \(\sinh{(x + y)} = \sinh{x} \cosh{y} + \cosh{x} \sinh{y}\). This formula resembles the addition formulas of sine and cosine in trigonometry, showing how hyperbolic functions can interrelate through such identities.

Understanding this formula helps in handling expressions involving multiple hyperbolic terms by transforming and reducing the complexity of computations. These transformations are invaluable in advanced calculus and physics, where systems often involve combinations of inputs and states.

• The formula confirms the distributive nature of hyperbolic functions over addition.
• It introduces a powerful tool for solving differential equations and evaluating integrals.

By mastering the addition formula, you’ll find it easier to engage with complex hyperbolic expressions, as it enables a seamless transition between different formats and arrangements of hyperbolic terms.