Question

Use the definitions of the hyperbolic functions to prove that each of the identities in Exercises 90–92 hold for all values of $x$and $y$. Note how similar these identities are to those which hold for trigonometric functions. (These exercises involve hyperbolic functions.)

$$\begin{gathered} \left(a\right)\sin h\left(-x\right)=-\sin hx \\ \left(b\right)\cos h\left(-x\right)=\cos hx \end{gathered}$$

Short answer

Part (a). We proved $\sin h\left(-x\right)=-\sin hx$.

Part (b). We proved$\cos h\left(-x\right)=\cos hx$.

Step-by-step solution

Part (a) Step 1. Given Information

Using the definitions of the hyperbolic functions we need to prove that$\sin h(-x)=-\sin hx,\cos h\left(-x\right)=\cos hx$.

Part (a) Step 2. Explanation

$$\begin{gathered} \sin h\left(-x\right) \\ =\frac{e^{-x}-e^{-\left(-x\right)}}{2} \\ =\frac{e^{-x}-e^x}{2} \\ =-\left(\frac{e^x-e^{-x}}{2}\right) \\ =-\sin hx \end{gathered}$$

Part (b) Step 1. Explanation 

$$\begin{gathered} \cos h\left(-x\right) \\ =\frac{e^{-x}+e^{-\left(-x\right)}}{2} \\ =\frac{e^{-x}+e^x}{2} \\ =\left(\frac{e^x+e^{-x}}{2}\right) \\ =\cos hx \end{gathered}$$