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.)

$\cos h\left(x+y\right)=\cos hx\cos hy+\sin hx\sin hy$

Short answer

We proved$\cos h\left(x+y\right)=\cos hx\cos hy+\sin hx\sin hy$using the definitions of the hyperbolic functions.

Step-by-step solution

Step 1. Given Information  

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

Step 2. Proving the statement 

$$\begin{gathered} \cos hx\cos hy+\sin hx\sin hy \\ =\left(\frac{e^x+e^{-x}}{2}\right)\left(\frac{e^y+e^{-y}}{2}\right)+\left(\frac{e^x-e^{-x}}{2}\right)\left(\frac{e^y-e^{-y}}{2}\right) \\ =\frac{e^x{e}^y+e^{-x}{e}^y+e^x{e}^{-y}+e^{-x}{e}^{-y}+e^x{e}^y-e^{-x}{e}^y-e^x{e}^{-y}+e^{-x}{e}^{-y}}{4} \\ =\frac{2e^x{e}^y+2e^{-x}{e}^{-y}}{4} \\ =\frac{e^{x+y}+e^{-\left(x+y\right)}}{2} \\ =\cos h\left(x+y\right) \end{gathered}$$