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{\bf Five More Proofs of the Cosine Addition Formula (Inspired by Mark Levi's Perpetuum Mobile Proof) }
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{\it Doron ZEILBERGER}

\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad {\it Dedicated to my hero Mark Levi}

Every time I get {\it SIAM  News}, I immediately turn to Mark Levi's wonderful column {\it Mathematical Curiosities} {\tt https://www.marklevimath.com/sinews}.
In the current issue, Levi [L] gave a {\it proof from the book} of the trigonometric identity $\cos(\alpha +\beta) \,= \, \cos \beta \cdot  \cos \alpha - \, \sin \beta \cdot  \sin \alpha$,
by showing that it follows from the non-existence of {\bf 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). 

{\bf Pre-calculus}: See  the Wikipedia article {\it List of Trigonometric Identities}. \halmos

{\bf Linear Algebra}: Rotating the vector $[1,0]^T$ by an angle of $\alpha$ gives the vector $[\cos \alpha, \sin \alpha]^T$.
Rotating the vector $[0,1]^T$ by an angle of $\alpha$ gives the vector $[-\sin\alpha, \cos \alpha]^T$. By {\bf linearity}, the
{\bf rotation  matrix}, $R_\alpha$ is
$$
R_{\alpha} = \left [ \matrix{\cos \alpha & -\sin \alpha \cr
                          \sin \alpha &  \cos \alpha}
          \right ] \quad .
$$
Now use {\bf matrix multiplication} and the fact that $R_{\alpha + \beta}=R_{\alpha} R_{\beta}$. \halmos

{\bf Complex Variable}: Take the real parts of  both sides of $e^{i(\alpha+\beta)}=e^{i\alpha} e^{i\beta}$  \halmos

{\bf Differential Equations}: Both sides satisfy the differential equation (viewed as a function of $\alpha$) $y''+y=0$, subject to the initial conditions $y(0)=\cos \beta$, $y'(0)=-\sin \beta$. Now use uniqueness. \halmos

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

{\bf References}

[L] Mark Levi, {\it Cosine Addition Formula and Perpetual Motion}, SIAM News {\bf 55 \#3} (April 2022), p.7.
%{\tt https://www.marklevimath.com/\_files/ugd/4d55eb\_d192c1ba72664caea4ca153344ac4ebb.pdf} \quad .

[Z] Doron Zeilberger, {\it Enumerative and Algebraic Combinatorics}, in: {\it ``Princeton Companion to Mathematics''} , (Timothy Gowers, ed.), Princeton University Press,2008, 550-561. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimPDF/enuPCM.pdf \quad .
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{\bf  D. Zeilberger, Mathematics Department, Rutgers University}, {\tt DoronZeil@gmail.com} \quad .




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