trigonometry problems (1 Viewer)

[JoeyH]

I can't do these problems on trigonometry, can someone help me on these please ?

Show that for all x which is an element of R,

cos x = 1- (1/2!)x^2 + (1/4!)x^2...and sin x = x - (1/3!)x^3 + (1/5!)x^5...​

(a) To do this, let C(x) and S(x) denote the sums of the two series. Verify that these sums converge for all x that is an element of R, and that C′′(x) = −C(x) and that S′′(x) = −S(x) for all x that is an element of R.

(b) It is a fact that the only functions f(x) satisfying f′′(x) = −f(x) for all x are the functions f(x) = Acos x + B sin x. Assuming this, show that C(x) = cos x and that S(x) = sin x. For example, for the function C(x), you must show that the A is 1
and that the B is 0.

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Suppose that f(x) is a twice differentiable function which satisfies f′′(x) = −f(x) for all x. We show that f(x) = Acos x + B sin x for certain constants A and B as follows:
(a) Show that f(x) cos x − f′(x) sin x is constant.
(b) Show that f(x) sin x + f′(x) cos x is constant.
(c) Use the first two parts to complete the solution.​

Your input will be very much appreciated.​