prove the derivative of sin is cos

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)

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