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Chapter 6: Problem 3

What is the fundamental identity for hyperbolic functions?

Short Answer

Expert verified
Answer: The fundamental identity for hyperbolic functions is cosh^2(x) - sinh^2(x) = 1, which relates the hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions.

Step by step solution

01

Define the hyperbolic sine and cosine functions

The hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions are defined as follows: sinh(x) = \frac{e^x - e^{-x}}{2} cosh(x) = \frac{e^x + e^{-x}}{2}
02

Square the sinh and cosh functions

To establish a relationship between these functions, let's first square both equations: sinh^2(x) = \left(\frac{e^x - e^{-x}}{2}\right)^2 cosh^2(x) = \left(\frac{e^x + e^{-x}}{2}\right)^2
03

Expand the squared functions

Next, we'll simplify both the squared equations by expanding. sinh^2(x) = \frac{e^{2x} - 2 + e^{-2x}}{4} cosh^2(x) = \frac{e^{2x} + 2 + e^{-2x}}{4}
04

Establish the fundamental identity

Now subtract sinh^2(x) from cosh^2(x): cosh^2(x) - sinh^2(x) = \frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4} Upon subtracting the two equations, we arrive at our fundamental identity for hyperbolic functions: cosh^2(x) - sinh^2(x) = 1

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