Def. Let x R. The function sin(x) (resp. cos(x)) is defined to be the y-coordinate (resp x-coordinate) of the point on the unit circle at an angle of x radians (measured counterclockwise from the positive x-axis).

(a) Prove that if f : (a, b) (c, d) is an increasing and surjective function, then it is continuous.

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

(c) Prove (without using L'Hospital's rule!), that limx0 [sin(x)/x] = 1 and limx0 ([cos(x) 1]/x) = 0. Note this implies sin(x) and cos(x) are both differentiable at 0.

(d) Prove that sin(x) = cos(x) and cos(x) = sin(x), using the angle sum formulas and previous steps.