This example verifies the inverse property of the exponential function using the distributive property of multiplication.
Start with the inverse property of the exponential function.
Multiply both sides by and rearrange the equation so is on the left side.
Apply the power series definition of the exponential function, shown below, to both instances of the exponentail function from above.
After substituting the power series definition we are left with the following expression.
Note, the expressions where is raised to an odd power will flip the sign to negative and expressions where is raised to an even power the sign will stay positive. This is shown below.
Then, we can use a geometric interpretation of the distributive property of multiplication to evaluate the product of the two series. This is represented below as the area of the rectangle.
This geometric interpretation of the multiplication is illustrated in the figure below. The area of the whole is equal to all of the sub-areas summed together.
The summation of the geometric interpretation of multiplication is shown below.
Group the same degree polynomials and evaluate the sum.
Since and all of the other polynomials sum to this verifies the inverse property.