Can you derive the result differently? When the solution
that we have finally obtained is long and involved,
we naturally suspect that there is some clearer and less
roundabout solution.  Can you derive the result differently?
Can you see it at a glance? Yet even if we have
succeeded in finding a satisfactory solution we may still be 
interested in finding another solution. We desire to convince
ourselves of the validity of a theoretical result by 
two different derivations as we desire to perceive a
material object through two different senses. Having found a 
proof, we wish to find another proof as we wish to touch 
an object after having seen it. Two proofs are better than one.
"it is safe riding at two anchors."
1.  Example Find the area S of the lateral surface of 
the frustum of a right circular cone, being given the 
radius of  the lower base R, the radius of the upper base r, 
and the altitude h.
     This problem can be solved by various procedures. For 
instance we may know the formula for the lateral surface 
of a full cone. As the fur frustum is generated by cutting off
form of cone a smaller cone, so its lateral surface is the
 difference of two conical surfaces; it remains to express
these terms R,r,h. Carrying through this idea, 
we obtain finally the formula
s=pi(R+r)sqrt(R-r)^2+h^2)
     Having found this result in some way or other, after
longer calculation, we may desire a clearer and less
roundabout argument. Can you derive the result differently?
Can you see it at a glance?  
     Desiring to see intuitively the whole result, we may 
begin with trying to see the geometric meaning of its
parts. Thus, we may observe that
     sqrt((R-r)^2+h^2)
     is the length of the  slant height (the slant height is one
of the non parallel sides of the isosceles t*****zoid that
revolving about the line joining the midpoints of its
parallel sides, generates the frustum; see fig 12.) Again, 
we may discover that
pi(R+r) = (2piR+2pir) / 2

is the arithmetic mean of the perimeters of the two bases
of the frustum. Looking at the the same part of the formula
we may be moved to write it also in the form

pi(R+r) = 2pi(R+r)/2

that is the perimeter of the midsection of the frustum.
(we call here mid-section the intersection of the frustum
with a plane which is parallel both to the lower base and
to the upper base of the frustum and bisects the altitude.)

                      r
          .-~*****====_. 
         .*=._._.C___-~*\
        /.       |        \
       / .     h |         \ 
      /  .       |          \
     /   .   ____L____       \
    /.--~.***    |    * R ~--.\
    ............L~~~~~~~~~~~~~
         ****~~~~~~~~~~****


Having found new interpretations of various parts, we 
may see now the whole formula in a different light. We
may read it thus;

Area = Perimeter of midsection * slant height.
 
We may recall here the rule for the t*****zoid;

Area = Middle line * altitude.

(the middle line is parallel to the two parallel sides of
the t*****zoid and bisects the altitude.) Seeing intuitively
the analogy of both statements, that about the frustum
and that about the t*****zoid=, we may see the whole result
about the frustum "almost at a glance."  That is, we feel
that we are very near now to a short and direct proof of
the result found by a long calculation.
2.  The foregoing example,l.e is typical.  Not entirely satisfied-
satisfied with our derivation of the result, we wish to improve
it, to change it. Therefor, we study the result, trying to 
understand it better, to see some new aspect  of it.  We
may succeed first in observing a new interpretation of a 
certain small part of the result.  Then, we may be lucky
enough to discover some new mode of conceiving some
other part.
   Examining the various parts, one after the other and 
trying various ways of considering them, we may be led
finally to see the whole result in a different light, and our
new conception of the result may suggest a new proof.
     It may be confessed that all this is more likely to hap-
pen to an experienced mathematician dealing with some
advanced problem than to a beginner struggling with
some elementary problem than to a beginner struggling with
some elementary problem. The mathematician who has a 
great deal of knowledge is more exposed than the 
beginner to the danger of mobilising too much knowledge and
framing and unnecessarily involved argument. But, as a 
compensation, the experienced mathematician is in a 
better position than the beginner to appreciate the 
reinterpretation of a small part of the result and to proceed, 
accumulating such small advantage, to recasting ultimately
the whole result.
     Nevertheless, it can happen even in very elementary 
classes that the students present an unnecessarily complicated
 solution. Then, the teacher should show them
at least once or twice, not only how to solve the problem
more shortly but also how to find, in the result itself,
indications of a shorter solution.
     See also REDUCTIO AD ABSURDUM AND INDIRECT PROOF