Bridge to Abstract Mathematics: Mathematical Proof and Structures

150 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5

that two quantities, say, A and 2, are equal, we have provided a string of

equations:

beginning with A and ending with 2. This form of proof, which we refer

to as a proof by transitivity, is the most elegant and desirable form in which

a conclusion whose statement involves V, but not 3 or +, can be written.

We give examples later of proofs by transitivity of conclusions asserting

the relationships = and I between real numbers, and = and r between

pairs of sets.

EXAMPLE 3 (Trigonometry) Derive the identity

(cos 2x - sin2 x)/sin 2x = *cot x - tan x,

for any real number x, not of the form x = n42, where n is an integer.

Use double angle formulas for sine and cosine and the definitions of cot

and tan, in terms of sin and cos.

Solution Let x be a real number, not of the form x = nn/2, where n is an

integer. Then,

(COS 2x - sin2 x)/sin 2x = [(cos2 x - sin2 x) - sin2 x]/sin 2x

= [(cos2 x - 2 sin2 x)]/sin 2x

= [(cos2 x - 2 sin2 x)]/2 sin x cos x

= ((COS x/2 sin x)) - (sin xlcos x)

=&cot x - tan x.

In this proof, studied at a more advanced level of high school mathematics

than the proof in Example 2, we have altered the approach taken in Example

2 by not stating explicitly a justification for each step. This is common in

proof writing past the elementary level. We note at the same time, however,

that the proof has been carefully laid out, with each line representing a step

that is clearly intelligible to those who are reasonably well-informed. If spe-

cific reasons for individual steps are not supplied in a proof (as is customary),

then the writer of a proof must use good judgment in providing a fair

amount of detail; with to^ few steps, it might not be possible to follow the

reasoning. Also, it is often appropriate to state explicitly the justifica-

tion for a step that is either especially important or particularly tricky to

understand.

On the other hand, we could have inserted into the proof of Example

3 (between lines 3 and 4) an additional line containing the expression

I

[(cos2 x)/(2 sin x cos x)] - [(2 sin2 x)/(2 sin x cos x)]

but chose not to do so. Surely most readers can follow the progression from

t line^3 to line 4 in the form given; with too many steps, the proof becomes