Two EZ Proofs of sin^{2}z + cos^{2}z = 1

Leon EHRENPREIS^{1} and Doron ZEILBERGER^{1}

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 = nπ or (n + \frac{1}{2})π , 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 π/2.

Now we appeal to a theorem on entire functions. f(z) is an entire function of exponential type 2. In fact |f(z)| ≤ Ce^{2|z|}. It is a standard consequence of the argument principle that such a function cannot have more than (2 + ϵ)r/π zeros (counting multiplicities) in |z| ≤ r, unless it vanishes identically. But we have produced 4r/π zeros there. Thus f(z) ≡ 0.

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 exponential generating function (henceforth e.g.f.)^{2} 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} < … < a_{r}; b_{1} < … < b_{s}) such that r + s is even, {a_{1}, …, a_{r}, b_{1}, …, b_{s}}= {1, 2, …, r + s}, and the weight is (−1)^{[r/2]+[s/2]}. Let the mate of such a pair be (a_{1}, …, a_{r}, b_{s}; b_{1}, …, b_{s−1}) if a_{r} < b_{s}, and (a_{1}, …, a_{r−1} ; b_{1}, …, 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. ^{3}

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 up-down permutations is given by sec z + tan z.

Nisan 12, 5754.

1 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. [leon,zeilberg]@math.temple.edu.

Supported in part by the NSF.

2 The e.g.f. of a combinatorial family of labelled objects according to a weight w is ∑_{n=0}^{∞} a_{n}z^{n}/n!, where a_{n}

is the sum of the weights of all the objects of size n. The product A×B of two such combinatorial families is the set of ordered pairs (a,b), a ∈ A , b∈ 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 × 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.

3 A similar (but not identical) proof, that will appear in Math. Magazine, was found independently by Ed Scheinerman.