What do the limits and have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?

Step 1 of 3:

In this problem, we need to explain what do the limits have to do with the derivatives of the sine and cosine functions and we have to find the derivatives of these functions.

Step 2 of 3:

We know that those limits are directly used in calculating the derivatives of sin x and cos x.

So, we firstly solve the derivative of sin x

If we denote we can write

…………..(1)

Now we solve

Clearly if we put the limit then we get indeterminate form.

So, we apply L-Hospital Rule

Then, we get

And

From equation (1)

for any real