Question

Prove the following identities. $$\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$$

Short answer

Question: Prove the identity involving hyperbolic sine and cosine functions: $$\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$$ Answer: Using the definitions of hyperbolic sine and cosine functions, we manipulated the terms in the equation until the left side equaled the right side of the given identity. The proof involves replacing functions with their definitions, distributing and simplifying terms, and cancelling out common expressions. The identity is proven as both sides of the equation are equal: $$\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$$

Step-by-step solution

Write down the definitions of hyperbolic sine and cosine functions

We will first write down the definitions of the hyperbolic sine and cosine functions: $$\sinh x = \frac{e^x - e^{-x}}{2}$$ $$\cosh x = \frac{e^x + e^{-x}}{2}$$

Replace the functions in the identity with their definitions

Replace the functions in the given identity with their definitions: $$\frac{e^{(x+y)} - e^{-(x+y)}}{2} = \frac{e^x - e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} + \frac{e^x + e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2}$$

Simplify the equation

Distribute terms on the right side of the equation and simplify: \begin{align*} \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}\\ &= \frac{e^{x+y} + e^{x-y} - e^{-x+y} - e^{-(x+y)} + e^{x+y} - e^{x-y} + e^{-x+y} - e^{-(x+y)}}{4} \end{align*}

Observe the equation and cancel out terms

Notice how some expressions in the equation cancel each other out: $$\frac{e^{(x+y)} - e^{-(x+y)}}{2} = \frac{2e^{x+y} - 2e^{-(x+y)}}{4}$$

Simplify further

Simplify the fraction by dividing both numerator and denominator by 2: $$\frac{e^{(x+y)} - e^{-(x+y)}}{2} = \frac{e^{x+y} - e^{-(x+y)}}{2}$$ Now, as we see, both sides of the equation are equal, and thus the identity is proven.

Key concepts

Hyperbolic Sine

The hyperbolic sine, denoted as \( \text{sinh} \), is one of the fundamental hyperbolic functions akin to the trigonometric functions but for a hyperbola, rather than a circle. The function is defined using exponential functions by the equation:
\[ \text{sinh}(x) = \frac{e^x - e^{-x}}{2} \]
This equation demonstrates the 'odd' nature of the hyperbolic sine function—that is, \( \text{sinh}(-x) = -\text{sinh}(x) \). This property becomes very useful in the mathematical analysis and modeling of systems that exhibit symmetry with respect to the origin or those that change direction with time.

Hyperbolic Cosine

Similarly, the hyperbolic cosine, \( \text{cosh} \), is another primary hyperbolic function related to the shape of a hyperbola.
It's defined as:
\[ \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \]
Contrary to \( \text{sinh} \), \( \text{cosh} \) is an 'even' function, satisfying \( \text{cosh}(-x) = \text{cosh}(x) \). The hyperbolic cosine represents the average of two exponential functions and often comes up in the study of hanging cables or bridges, known as catenaries, as well as in other areas of physics and engineering.

Exponential Functions

Exponential functions are mathematical expressions involving an exponent that contains a variable. They are written in the form \( e^x \) where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Exponential functions exhibit a rapid rate of growth and are omnipresent in many areas, including compound interests in finance, population growth in biology, and decay of radioactive elements. In calculus, they are unique as their derivative is proportional to the function itself, making them an essential part of differential equations and complex analysis.

Identity Verification

Identity verification in mathematics pertains to proving that two expressions represent the same object under all circumstances within their domain.
For hyperbolic functions, verifying an identity such as \( \text{sinh}(x+y) = \text{sinh} x \text{cosh} y + \text{cosh} x \text{sinh} y \) requires substituting the functions with their exponential definitions and then simplifying the equation through algebraic manipulation. This process often involves expanding products, canceling out terms, and reducing fractions to reach a point where both sides of the equation match, thus confirming the identity as true.