Chapter 2: Differentiation

Section 2.9: The Hyperbolic Functions and Their Derivatives

Example 2.9.3

$$ Verify the first hyperbolic Pythagorean identity in Table 2.9.5. $$ Solution $$ As in Example 2.9.1, verification of the (hyperbolic) Pythagorean, addition, and double-angle identities is effected by replacing each hyperbolic function with its exponential equivalent. Think of this approach as the eeezy way. Verification of the Pythagorean identity is as follows.   ${\cosh }^2\left(x\right)-{\sinh }^2\left(x\right)$ $\={\left(\frac{e^x\+e^{-x}}{2}\right)}^2-{\left(\frac{e^x-e^{-x}}{2}\right)}^2$ $$ $\=\frac{1}{4}\left(e^{2 x}\+2e^{x-x}\+e^{-2 x}\right)-\frac{1}{4}\left(e^{2 x}-2e^{x-x}\+e^{-2 x}\right)$ $$ $\=\frac{1}{4}\left(e^{2 x}-e^{2 x}\+2e^0\+2e^0\+e^{-2 x}-e^{-2 x}\right)$ $$ $\=\frac{1}{4}\left(2\+2\right)$ $$ $\=1$ $$

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