Using elementary geometry and the definition of sin x, cos x, we can show that for every x...

Using elementary geometry and the definition of sin x, cos x, we can show that for every x, y ∈ R (see Appendix B)

i)

ii)

iii)

iv)

Moreover, if x is measured in radians, then

v)

and

vi)

Using these properties, prove each of the following statements.

a) The functions sin x and cos x are continuous at 0.

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

c) The limits

exist.

d) The function sin x is differentiable on R with (sin x)′ = cos x.

e) The functions cos x and tan x := sin x/ cos x are differentiable on R with (cos x)′ = −sin x and (tan x)′ = sec2 x.