One way to define sin and cos is as follows: sin(x)=(-1)", n=0 and 22n+1 (2n + 1)! 00 cos(x)=(-1)" n=0 x2n (2n)! (a) Prove that these power series converge everywhere. (b) Prove that sin'(x) = cos(x) and cos'(x) = sin(x). (e) Determine the zeros of sin and cos. (c) Prove that sin(x) + cos(x) = 1 for all x R. (d) Show that sin and cos are 27 periodic. (f) Prove that sin is not a rational function; i.e., it cannot be written as P(x)/Q(x) for polynomials P and Q. (g) Prove that sin is not even defined implicitly by an algebraic equation; that is, there is no rational functions fo, fn-1 such that (sin x)" + fn-1(x) (sin x)-1++fo(x) = 0.