Question

Verify the following identities. \(\cosh \left(\sinh ^{-1} x\right)=\sqrt{x^{2}+1},\) for all \(x\)

Short answer

Question: Verify the identity \(\cosh\left(\sinh^{-1}x\right) = \sqrt{x^2 + 1}\). Answer: The identity is verified by following the step-by-step solution outlined above, which involves recalling the definitions of inverse hyperbolic sine and hyperbolic cosine, substituting these definitions into the identity, and simplifying the expressions using the properties of exponentials, logarithms, and common denominators. The final expression obtained is \(\sqrt{x^2 + 1}\), confirming the validity of the given identity.

Step-by-step solution

Recall the definition of \(\sinh^{-1}x\)

First, we will recall the definition of the inverse hyperbolic sine function: $$\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})$$

Substitute the definition of \(\sinh^{-1}x\) into the identity

Now, substitute the expression from step 1 into the given identity: $$\cosh\left(\sinh^{-1}x\right) = \cosh\left(\ln(x + \sqrt{x^2 + 1})\right)$$

Recall the definition of the hyperbolic cosine function

Next, we will recall the definition of the hyperbolic cosine function: $$\cosh y = \frac{e^y + e^{-y}}{2}$$

Apply the definition of the hyperbolic cosine function to the identity

Using the definition of the hyperbolic cosine function from step 3, substitute \(y = \ln(x + \sqrt{x^2 + 1})\) into the identity: $$\cosh\left(\ln(x + \sqrt{x^2 + 1})\right) = \frac{e^{\ln(x + \sqrt{x^2 + 1})} + e^{-\ln(x + \sqrt{x^2 + 1})}}{2}$$

Simplify the expression using the properties of exponentials and logarithms

Using the properties of exponentials and logarithms, simplify the expression in the identity: $$\frac{e^{\ln(x + \sqrt{x^2 + 1})} + e^{-\ln(x + \sqrt{x^2 + 1})}}{2} = \frac{x+\sqrt{x^2+1}+\frac{1}{x+\sqrt{x^2+1}}}{2}$$

Simplify the expression by finding a common denominator

To further simplify the expression, find a common denominator for the two terms in the numerator: $$\frac{x+\sqrt{x^2+1}+\frac{1}{x+\sqrt{x^2+1}}}{2} = \frac{(x+\sqrt{x^2+1})^2 + 1}{2(x+\sqrt{x^2+1})}$$

Simplify the numerator and the denominator

Now, simplify the numerator and the denominator: $$\frac{(x+\sqrt{x^2+1})^2 + 1}{2(x+\sqrt{x^2+1})} = \frac{x^2+2x\sqrt{x^2+1}+x^2+1+1}{2(x+\sqrt{x^2+1})} = \frac{2x^2+2x\sqrt{x^2+1}+2}{2(x+\sqrt{x^2+1})}$$

Cancel out the common factors in the numerator and denominator

Cancel out the common factors (2) in the numerator and denominator: $$\frac{2x^2+2x\sqrt{x^2+1}+2}{2(x+\sqrt{x^2+1})} = \frac{x^2+x\sqrt{x^2+1}+1}{x+\sqrt{x^2+1}} = \sqrt{x^2 + 1}$$ Thus, we have verified that \(\cosh\left(\sinh^{-1}x\right)=\sqrt{x^2+1}\) for all \(x\).

Key concepts

Hyperbolic Sine Function

The hyperbolic sine function, denoted as \(\sinh x\), is a fundamental hyperbolic function similar to the trigonometric sine function, but it relates to hyperbolas rather than circles. The hyperbolic sine of \(x\) can be defined using exponential functions as follows:
\[\sinh x = \frac{e^x - e^{-x}}{2}\]
This shows that \(\sinh x\) is constructed by taking the difference between the exponential of \(x\) and the negative exponential of \(x\), divided by 2. One of the key properties of the hyperbolic sine function is that it relates to the inverse hyperbolic function, \(\sinh^{-1}x\), which is also called the area hyperbolic sine. The inverse hyperbolic sine function, \(\sinh^{-1}x\), allows us to obtain \(x\) from \(\sinh x\) through a logarithmic formula:
\[\sinh^{-1}{x} = \ln(x + \sqrt{x^2 + 1})\]
This important identity helps us convert back and forth between the hyperbolic sine and its inverse, providing a useful tool when dealing with problems involving hyperbolic functions.

Hyperbolic Cosine Function

The hyperbolic cosine function, \(\cosh x\), is yet another important function in hyperbolic trigonometry. It is analogous to the cosine function in traditional trigonometry, but operates over a hyperbola. The mathematical definition of the hyperbolic cosine function is straightforward:
\[\cosh x = \frac{e^x + e^{-x}}{2}\]
Just like with the hyperbolic sine function, \(\cosh x\) is derived from exponential functions. However, here we take the sum of the exponential of \(x\) and the negative exponential of \(x\), divided by 2. The hyperbolic cosine has several interesting properties, including its evenness, which means \(\cosh(-x) = \cosh(x)\). This property is useful for simplifying expressions, especially when working with inverse functions. For instance, in the context of solving \(\cosh(\sinh^{-1} x)\), the identity expresses a complex logarithmic and exponential relationship, revealing that it simplifies elegantly to \(\sqrt{x^2+1}\), as demonstrated through detailed algebraic manipulations.

Exponential and Logarithmic Properties

Understanding the properties of exponential and logarithmic functions is crucial when dealing with hyperbolic functions. Exponentials, denoted as \(e^x\), have unique properties, such as \(e^{a+b} = e^a \cdot e^b\) and \(e^{a-b} = \frac{e^a}{e^b}\), that simplify complex calculations. When these properties are applied in the realm of hyperbolic functions, they allow us to manipulate and simplify seemingly complicated expressions.

Logarithms, the inverse operations of exponentials, are equally important. With properties like \(\ln(a \cdot b) = \ln a + \ln b\) and \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\), logarithms can transform multiplicative processes into additive ones, facilitating easier computation. These properties are essential when working with inverse hyperbolic functions, such as \(\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})\). By leveraging these properties, we successfully verify and solve identities, like transforming \(\cosh(\sinh^{-1}x)\) into simpler expressions. Understanding these principles of exponentials and logarithms enhances our ability to grasp how hyperbolic functions operate, enabling the resolution of complex mathematical problems with ease.