Five More Proofs of the Cosine Addition Formula (Inspired by Mark Levi’s Perpetuum Mobile Proof)

Doron ZEILBERGER

Dedicated to my hero Mark Levi

Every time I get SIAM News, I immediately turn to Mark Levi’s wonderful column Mathematical Curiosities https://www.marklevimath.com/sinews. In the current issue, Levi [L] gave a proof from the book of the trigonometric identity cos(α + β) = cos β · cos α − sin β · sin α, by showing that it follows from the non-existence of Perpetual Motion. This reminded me of five other proofs, none of them, admittedly, as nice as Mark Levi’s proof (although the last one is a close second).

Pre-calculus: See the Wikipedia article List of Trigonometric Identities.

Linear Algebra: Rotating the vector [1, 0]^{T} by an angle of α gives the vector [cos α, sin α]^{T}. Rotating the vector [0, 1]^{T} by an angle of α gives the vector [− sin α, cos α]^{T}. By linearity, the rotation matrix, R_{α} is

R_{α} = [ \begin{matrix}cos α & − sin α \\ sin α & cos α \end{matrix} ] .

Now use matrix multiplication and the fact that R_{α+β} = R_{α}R_{β}.

Complex Variable: Take the real parts of both sides of e^{i(α+β)} = e^{iα}e^{iβ}

Differential Equations: Both sides satisfy the differential equation (viewed as a function of α) y′′ + y = 0, subject to the initial conditions y(0) = cos β, y′(0) = − sin β. Now use uniqueness.

Combinatorics: cos α (resp. sin α) is the exponential generating function of increasing sequences of integers of even length (resp. of odd length) with weight (−1)^{length/2} (resp. (−1)^{(length−1)/2}), see [Z]. Hence cos(α + β) is the exponential generating function of two-colored increasing sequences of even length, with say, colors α and β. If the number of integers colored α is even (resp. odd) we get the first term (resp. second term) on the right.

References

[L] Mark Levi, Cosine Addition Formula and Perpetual Motion, SIAM News 55 #3 (April 2022), p.7.

[Z] Doron Zeilberger, Enumerative and Algebraic Combinatorics, in: “Princeton Companion to Mathematics” , (Timothy Gowers, ed.), Princeton University Press,2008, 550-561.

https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enuPCM.pdf   .

D. Zeilberger, Mathematics Department, Rutgers University, DoronZeil@gmail.com; April 17, 2022. Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger.