One way to define sin and cos is as follows: sin(x)=(-1), n=0 and 22n+1 (2n +...

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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.¹ (b) Prove that sin'(x) = cos(x) and cos'(x) = sin(x). (e) Determine the zeros of sin and cos. (c) Prove that sin²(x) + cos²(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. 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.¹ (b) Prove that sin'(x) = cos(x) and cos'(x) = sin(x). (e) Determine the zeros of sin and cos. (c) Prove that sin²(x) + cos²(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.

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a We need to show that the series 2 2n1 sinx 1 2n 1 and 10 2 2n cosx 1 2n both converge for all x We will do this by showing that the terms in each se... View the full answer