1. The algebra we go through in the limit calculations for the derivative of either sin(x) or cos(x), using the angle-addition formula and rearranging, accomplishes one thing: reducing these limits to the computation of the special limits sin(h)/h or (cos(h)-1)/h as h goes to 0.

2. Those two limits are just exactly the limits you would compute to evaluate the particular values sin'(0) or cos'(0).

3. If we know anything about the cosine function, it’s that it has a maximum at 0. Thus cos'(0) must be zero, without doing any calculation.

4. Thus we can complete the differentiation of the cosine function entirely without even the difficulty of going through the geometric arguments involving triangles inscribed in circles.

Kind of neat, I thought! Also it was nice to realize, and point out, the common mathematical theme of reducing a general case to an easier-to-solve particular case.

]]>I have to agree and disagree with you. I agree creativity is important – and schools are not great at creative math ….. So my very frustration, led me to the exciting world of MOOC’s – through open learning, I studied maths at Stanford, QUT, Kyoto Uni and more.

Without MOOC’s, I would be constrained in exploring the world of math; MOOC’s allow me to appreciate maths in a more creative way. ]]>

Note, there seems to be a typo on your line for lim h-> 0 sin h/h ]]>