Question

Verify that the given equations are identities. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

Short answer

Yes, the given equation is a hyperbolic identity: \( \cosh 2x = \cosh^2 x + \sinh^2 x\).

Step-by-step solution

Review the Identity Formula

The identity to verify is given by the equation \(\cosh 2x = \cosh^2 x + \sinh^2 x \).We need to show that this equation holds true for all values of \(x\) by using known hyperbolic identities.

Use Hyperbolic Identity

Recall the hyperbolic identity \(\cosh^2 x - \sinh^2 x = 1 \). This formula is crucial as it's similar to the Pythagorean identity for trigonometric functions.

Break Down \(\cosh 2x\)

Using another hyperbolic identity, \(\cosh 2x = \cosh^2 x + \sinh^2 x\), we see that it matches the right side of the given equation.The provided equation is thus a well-known hyperbolic identity and holds without further manipulation necessary.

Validate with Expanding

To confirm, you can use the definitions: \(\cosh x = \frac{e^x + e^{-x}}{2}\) and \(\sinh x = \frac{e^x - e^{-x}}{2}\).Upon expanding the left side using \(\cosh 2x = \frac{e^{2x} + e^{-2x}}{2}\), and realizing that the right side equals this too after simplifying,we further support that \(\cosh 2x = \cosh^2 x + \sinh^2 x\) holds.

Key concepts

Cosh Function

The cosh function, or hyperbolic cosine, is a fundamental component in hyperbolic identities. Essentially, \(\cosh x\) is defined as the average of exponential functions: \[\cosh x = \frac{e^x + e^{-x}}{2}\]. This function grows exponentially, both positively and negatively, as \(x\) increases or decreases. Its shape is similar to a parabola, but it follows the hyperbolic plane.

An important property of \(\cosh x\) is its evenness, meaning that \(\cosh(-x) = \cosh x\), which plays an integral part in forming identities and simplifying equations. Understanding the intrinsic nature of the cosh function helps in verifying identities like the given exercise.

Sinh Function

The sinh function, or hyperbolic sine, displays peculiar properties that are used widely in calculus and algebra. Its definition is: \[\sinh x = \frac{e^x - e^{-x}}{2}\]. This function is odd, meaning \(\sinh(-x) = -\sinh x\), making it distinct compared to \(\cosh x\).

In terms of growth, \(\sinh x\) grows positively as \(x\) increases and negatively as \(x\) decreases. Together with \(\cosh x\), it forms crucial hyperbolic identities such as \(\cosh^2 x - \sinh^2 x = 1\), which is a variant of the Pythagorean identity used for trigonometric functions.Understanding \(\sinh x\) is essential for both verifying and manipulating identities.

Pythagorean Identity

The Pythagorean identity in hyperbolic terms is given by \(\cosh^2 x - \sinh^2 x = 1\). This mirrors the trigonometric Pythagorean identity but for hyperbolic functions instead of circles.

Hyperbolic identities take form differently but serve similar purposes: they help relate hyperbolic functions to one another and simplify expressions to verify equations. For instance, the identity in the exercise: \(\cosh 2x = \cosh^2 x + \sinh^2 x\) would be confusing without understanding how to relate \(\cosh\) and \(\sinh\) under specific formulas.Thus, learning the Pythagorean identity for hyperbolic functions is critical as it forms a cornerstone in drawing relationships among various hyperbolic equations.

Verification of Identities

Verification of identities involves checking whether given equations hold true for all values within their domains. In hyperbolic functions, this often involves substitution and manipulation of known identities. For example, in the exercise, verifying \(\cosh 2x = \cosh^2 x + \sinh^2 x\) involves recognizing it as a known hyperbolic identity.

By using the foundational identity, \(\cosh^2 x - \sinh^2 x = 1\), and the definition of \(\cosh\) and \(\sinh\) in terms of exponentials, inconsistencies (if any) in the equation become apparent. Checking each function and breaking them down using exponential functions can further delineate if an equation stands as an identity.Through substitution and expansion, identities can be verified, ensuring confidence in complex mathematical problem-solving.