Verify the following using the definitions of the hyperbolic functions.
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.
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.