This example proves of the derivative of the sine function shown below.
Start by applying the limit definition of the derivative to the sine function.
Expand the expression using the sum of two angles identity.
Split the fraction in two.
Split the limit in two.
Factor the expression which is not related to out of the first limit. Factor out which is not related to out of the second limit.
Multiply the second limit by in the form to flip signs.
Then take both limits.
Substitute the values of the limits into the expression.
Simplify the expression.
This proves the derivative of sine.