No, it is close to 1 "when delta x is 0.0001 or some other small value near 0". The crucial point is that it keeps getting closer and closer to 1 as delta x gets closer to 0.
Whether that is valid or not depends upon exactly how you have proved that the derivative of sin(x) is cos(x). Most textbooks use the limit . If you already have that then the given limit follows immediately by letting .