(1) We know \cos x \leq 1 for all x. By considering the derivative of the function f(x) = x - \sin x prove that \sin x \leq x for x \geq 0.
(2) By considering the derivative of the function f(x) = \cos x - \left(1 - {x^2\over 2}\right) prove that \cos x \geq 1 - {x^2\over 2} for x \geq 0.
(3) By considering the derivative of the function f(x) = \left(x - {x^3 \over 3!}\right) - \sin x prove that \sin x \geq (x - {x^3 \over 3!}) for x \geq 0.
(4) By considering the derivative of the function f(x) = \cos x - \left(1 - {x^2 \over 2!} + {x^4\over 4!}\right) prove that \cos x \leq \left(1 - {x^2\over 2!} + {x^4 \over 4!}\right) for x \geq 0.
(5) What can you say about continuing this process?