# Derivative Proof of sin(x)

We can prove the derivative of sin(x) using the limit definition and the double angle formula for trigonometric functions.

## Derivative proof of sin(x)

For this proof, we can use the limit definition of the derivative.

Limit Definition for sin:

Using angle sum identity, we get

Rearrange the limit so that the sin(x)'s are next to each other

Factor out a sin from the quantity on the right

Seperate the two quantities and put the functions with x in front of the limit (We are only concerned with the limit of h)

We can see that the first limit converges to 1

and the second limit converges to 0.

We can plug in 1 and 0 for the limits and get cos(x)