5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR 4. 151
cluttered and boring. There is no universally correct answer; what con-
stitutes a proof that is both informative and pleasing to read is largely a
matter of individual judgment and taste, both of which generally improve
with experience in writing proofs.
Thus far, in this article, we have focused on how a proof is to be presented.
But knowing how to present a proof is of little value without being able
to discover it-to find a path linking A to 2-in the first place. The next
example highlights this issue.
EXAMPLE 4 Prove that (1 + sin x)/cot2 x = sin x/(csc x - I), for all values
of x where both quantities are defined.
Solution Note first that the statement to be proved involves the universal
quantifier ("for all") but no existential quantifier or implication arrow.
Consequently, we wish to write a proof by transitivity, as we did in
Examples 2 and 3. Since both expressions are rather complex, however,
it is probably not easy to find a direct route from the expression on the
left to that on the right. In such a situation the best procedure is to try
to change the form of both expressions, in the hope of reducing (or ex-
panding) both to a common third expression. Taking this approach,
and assuming that x is a real number for which both quantities we're
working with are defined, we find that
(1 + sin x)/cot2 x = (1 + sin x)(tan2 x)
= (1 + sin x)(sin2 x/cos2 x)
= (sin2 x)(l + sin x)/(cos2 x)
whereas
sin x/(csc x - 1) = sin x/((l/sin x) - 1)
= sin x/((l - sin x)/sin x)
= sin2 x/(l - sin x)
= (sin2 x)(l + sin x)/(l - sin x)(l + sin x)
= (sin2 x)(l + sin x)/(l - sin2 x)
= (sin2 x)(l + sin x)/(cos2 x)
At this point we have essentially solved the problem, but the question
remains how to present the proof. The easiest way, from the point of
view of the writer, is to append to the previous arguments the conclusion
(1 + sin x)/cot2 x = (sin2 x)(l + sin x)/(cos2 x)
= sin x/(csc x - 1)
whenever these expressions are all defined, so that the desired equality
is proved. The way to present the proof, however, is to combine
the two strings of equations into a single string, that is, to write a proof