Surprising identities / equations What are some surprising equations/identities that you have seen, which you would not have expected?

This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.

I'd request to avoid 'standard' / well-known results like $ e^{i \pi} + 1 = 0$.

Please write a single identity (or group of identities) in each answer.

I found this list of Funny identities, in which there is some overlap.
This one by Ramanujan gives me the goosebumps:

$$
\frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{ (4k)! (1103+26390k) }{ (k!)^4 396^{4k} } = \frac1{\pi}.
$$



P.S. Just to make this more intriguing, define the fundamental unit $U_{29} = \frac{5+\sqrt{29}}{2}$ and fundamental solutions to Pell equations,

$$\big(U_{29}\big)^3=70+13\sqrt{29},\quad \text{thus}\;\;\color{blue}{70}^2-29\cdot\color{blue}{13}^2=-1$$

$$\big(U_{29}\big)^6=9801+1820\sqrt{29},\quad \text{thus}\;\;\color{blue}{9801}^2-29\cdot1820^2=1$$

$$2^6\left(\big(U_{29}\big)^6+\big(U_{29}\big)^{-6}\right)^2 =\color{blue}{396^4}$$

then we can see those integers all over the formula as,

$$\frac{2 \sqrt 2}{\color{blue}{9801}} \sum_{k=0}^\infty \frac{(4k)!}{k!^4} \frac{29\cdot\color{blue}{70\cdot13}\,k+1103}{\color{blue}{(396^4)}^k} =  \frac{1}{\pi} $$

Nice, eh?
Jean Nicod's first single axiom DDpDqrDDtDttDDsqDDpsDps, where D stands for the Sheffer stroke ("NAND"), and we write in Polish notation.  Under the rule {D$\alpha$D$\beta$$\gamma$, $\alpha$} $\vdash$ $\gamma$, all tautologies (containing just "D" and variables) in classical logic can get derived.

CCCpqrCCrpCsp.  This is Lukasiewicz's 13 letter axiom which suffices as the only axiom, under detachment, to prove all formulas of the classical pure implicational calculus of propositions.

Meredith's 21 letter axiom CCCCCpqCNrNsrtCCtpCsp which suffices as the only axiom for C-N classical logic under detachment (it's not known if this is the shortest such axiom).

(x(y(((zz')(uy)')x))') = u.  This is a product and inverse single axiom for group theory.  There are no shorter product and inverse single axioms for group theory.

((x|z)|y)|((x|(x|y))|x)=y, (x|((y|x)|x))|(y|(z|x))=y, ((x|y)|z)|(x|((x|z)|x))=z, and  ((x|(z|x))|x)|(z|(y|x))=z are among the shortest possible single axiom for Boolean algebra in the Sheffer stroke.

XCB: EpEEEpqErqr was the last proved shortest single axiom and completes the set of shortest single axioms for equivalential calculus under the rule {E$\alpha$$\beta$\, $\alpha$ $\vdash$ $\beta$}.  The proof involved a computer program.

C$\delta$pC$\delta$Np$\delta$q, where $\delta$ comes as a functional variable of one argument, consists of a single axiom which under uniform substitution and detachment comes as sufficient to deduce all C-N-$\delta$ tautologies.

C$\delta$CpqCp$\delta$q suffices as a single axiom to prove a deduction theorem for a propositional calculus.

C$\delta$CpqC$\delta$p$\delta$q, from what I have read though I have not seen a proof, suffices as a single axiom for all C-$\delta$ propositions.

C$\delta$$\delta$0$\delta$p.  This 6 letter single axiom under detachment, substitution, and the rules for quantifiers, I have read, suffices for all theorems or axioms of classical propositional logic with functional variables, quantifiers of propositions, and quantifiers of functional variables.