56 CHAPTER 2. FUNCTIONS

Finally, the graph of x → sec(x) is of the form

- - /2 0 /2 3 /2 2 5 /2 3-4

-2

0

2

4sec(x)

Both of these functions have vertical asymptotes at odd multiples of π/2 although Ihave not shown them with the secant function.

The formula for the cosine and sine of the sum of two angles is also important. Likemost of this material, I assume the reader has seen it. However, I am aware that manypeople do not see these extremely important formulas, or if they do, they often see noexplanation for them so I shall give a review of it here.

The following theorem is the fundamental identity from which all the major trig. iden-tities involving sums and differences of angles are derived.

Theorem 2.3.5 Let x,y ∈ R. Then

cos(x+ y)cos(x)+ sin(x+ y)sin(x) = cos(y) . (2.4)

Proof: Recall that for a real number z, there is a unique point p(z) on the unit circleand the coordinates of this point are cosz and sinz. Now it seems geometrically clear thatthe length of the arc between p(x+ y) and p(x) has the same length as the arc betweenp(y) and p(0) . As in the following picture.

p(y)

p(x+ y)

p(x)

(1,0)