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

Question:

Using elementary geometry and the definition of sin x, cos x, we can show that for every x ˆŠ R
i) | sin x| ii) sin(-x) = - sin x. cos(-x) = cos x,
iii) sin2 x + cos2 x = 1, cos x = 1 - 2 sin2 (x/2),
iv) sin(x ± y) = sin x cos y ± cos x sin y. Moreover, if x is measured in radians, then v) cos x = sin (Ï€/2 - x), sin x = cos (Ï€/2 - x), and
vi) 0 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.