Using these properties and the Chain Rule (see p. 92), prove that each of the following statements.

(a) The functions sinx and cosx are continuous at O.

(b) The functions sin x and cos x are continuous on R.

(c) The limits exist.

1I· msi-n -x = 1 x--+O X and 1· 1- cos x 0 1m =

x--+O X

(d) The function sinx is differentiable on R with (sin x)' = cosx.

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(e) The functions cosx and tanx := sin x/ cos x are differentiable on R with

(cos x)' = - sin x and (tanx)' = sec2 x.