Verify the following using the definitions of the hyperbolic functions.

Question

csh x + sinh x = e^x

First, substitute the definitions of cosh x and sinh x. choose the correct substitution.

Then combine the fractions in the result above and simplify the numerator.

Then, simplify the rational expression above.

Answer 1

We start by substituting the definitions of cosh x and sinh x:

cosh x + sinh x = (e^x + e^(-x))/2 + (e^x - e^(-x))/2.

Combining the fractions, we get:

cosh x + sinh x = (e^x + e^(-x) + e^x - e^(-x))/2.

Simplifying the numerator, we have:

cosh x + sinh x = (2e^x)/2.

Simplifying the rational expression, we get:

cosh x + sinh x = e^x.