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\qquad \quad \qquad \qquad \qquad \qquad 
Two EZ Proofs of 
${\bf sin^2 z + cos^2 z =1}$
\medskip
\it
\qquad\qquad\qquad\qquad\qquad Leon EHRENPREIS$^1$ 
and Doron ZEILBERGER\footnote{$^1$}
{\eightrm 
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
{\eighttt [leon,zeilberg]@math.temple.edu.} Supported in part by the NSF.
}
\medskip
\rm
The two proofs contrast two origins of the sine function:
(a)The zeros of $\sin z$, and (b)The power series for $\sin z$.
 
For the proof based on (a), we study the zeros of 
$f(z):= \sin ^2 z + \cos ^2 z -1$. Observe that $f$ vanishes when
$z= n \pi$ or $(n+ {{1} \over {2}} ) \pi$ , for any integer $n$. 
This is the Pythagorean
theorem for degenerate triangles. But, even more, $f$ has a second order
zero at these points, because $f$ is even (so it must have a double
zero at $z=0$), and periodic of period $\pi /2$.
 
Now we appeal to a theorem on entire functions. $f(z)$ is an entire function
of exponential type $2$. In fact $|f(z)| \leq C e^{2 |z|}$. It is
a standard consequence of the argument principle that such a function cannot
have more than $(2+ \epsilon ) r/ \pi$ zeros (counting multiplicities) in
$|z| \leq r$, unless it vanishes identically. But we have produced
$4r / \pi $ zeros there. Thus $f(z) \equiv 0$. \halmos

The same method of proof can be used to prove many identities
for elliptic functions.
 
 
For the proof based on (b),  observe that $\sin z$ (resp. $\cos z$)
is the {\it exponential generating function} 
(henceforth e.g.f.)\footnote{$^2$}
{\eightrm The e.g.f. of a combinatorial family of labelled objects
according to a weight $w$ is $\sum_{n=0}^{\infty} a_n z^n / n!$, where
$a_n$ is the sum of the weights of all the objects of
size $n$. The product A$\times$B of two such combinatorial families
is the set of ordered pairs (a,b), a $\in$ A , b$\in$ B,
with the labels of a and b disjoint, size(a,b):=size(a)+size(b),
and weight(a,b)=weight(a)weight(b). It is easy to see that the
e.g.f. of A $\times$ B is the products of the e.g.f.s of A and B.
See  D.Foata, M.Schutzenberger, LNM \# 138 (Springer), and H. Wilf,
``generatingfunctionlogy'', Academic Press.}
for increasing sequences of integers
of odd (resp. even) size, weighted by $(-1)^{[size/2]}$.
Hence $\sin^2 z + \cos^2 z$ is the e.g.f. for ordered pairs of
increasing sequences of integers 
$( a_1 < \dots < a_r \, ;\,  b_1 < \dots < b_s )$
such that $r+s$ is even, $\{ a_1 , \dots , a_r , b_1 , \dots , b_s \}$=
$\{1,2, \dots , r+s \}$, and the weight is 
$(-1)^{[r/2] + [s/2]}$. Let the {\it mate} of such
a pair be $( a_1 , \dots , a_r , b_s \, ; \, b_1 , \dots , b_{s-1} )$ if
$a_r < b_s$, and $( a_1 , \dots , a_{r-1} \, ; \, b_1 , \dots , b_{s}, a_r )$ 
if $a_r > b_s$. Since every pair has opposite sign from its mate, the total
weight of each couple is $0$. The only left-over is the pair $(empty,empty)$
that has no mate; Its weight is $1$, and its size is $0$, hence
its $e.g.f.$ is $1$. \halmos\footnote{$^3$}
{\eightrm A similar (but not identical) 
proof, that will appear in Math. Magazine, 
 was found independently by Ed Scheinerman.}

The same method of proof can be used to prove many other trig. identities,
and also Andre's result that the e.g.f of {\it up-down permutations} is given
by $ \sec z + \tan z$.

 
{\bf Nisan 12, 5754.}
\bye
 
 
 

