The functional equation identity, (assuming ALSO $\,f(-x)=-f(x)\,$ for all $\,x$),

$$ f(a)f(b)f(a\!-\!b)+f(b)f(c)f(b\!-\!c)+f(c)f(a)f(c\!-\!a)+ f(a\!-\!b)f(b\!-\!c)f(c\!-\!a) = 0 \tag{1}$$

for all $\,a,b,c\,$ has solutions $f(x)=k_1\sin(k_2\,x)$ and $f(x)=k_1\tan(k_2\,x)\,$ with $\,k_1,k_2\,$ complex constants.

As a limiting case of both $\,\sin\,$ and $\,\tan,\,$ $\,f(x)=k_1x\,$ is also a solution, and the simplest.

I am looking only for non-zero solutions that have a formal power series expansion. That is, $$ f(x) = a_1 x^1/1! + a_3 x^3/3! + a_5 x^5/5! + \cdots \tag{2}$$ is the exponential generating function for the sequence $\,(0,a_1,0,a_3,0,a_5,0,\dots).\,$ For a solution of the above functional equation, if $\,a_1=0\,$ then $\,f(x)\equiv0.\,$ Otherwise, $\,a_1\ne 0\,$ and $\,a_3,a_5\,$ can be arbitrary and the rest of the coefficients are determined uniquely. I used Mathematica to compute the first few coefficients. I found, for example, $$ a_7 \!=\! \frac{11 a_1 a_3 a_5\!-\!10 a_3^3}{a_1^2}, a_9 \!=\! \frac{21 a_1^2 a_5^2 \!+\!60 a_1 a_3^2 a_5 \!-\!80 a_3^4}{a_1^3}, \dots. \tag{3}$$

I know of 18 identities for sin and tan similar to this one. They have common features as follows.

Each is an irreducible homogeneous polynomial equated to zero where each monomial term in the polynomial is a product of factors each of which is of the form $\,f(x)\,$ where $\,x\,$ is a variable or an integer linear combination of variables. I also require that $\,f(x) = k_1x\,$ is a solution in which case the functional equation is a homogeneous algebraic identity.

As a **non-example**, the similar looking
**non-homogeneous** functional equation
$$ f(a\!-\!b)\!+\!f(b\!-\!c)\!+\!f(c\!-\!a)\!-\!
f(a\!-\!b)f(b\!-\!c)f(c\!-\!a)\!=\!0 \tag{4}$$
has only the **non-zero**
solutions $\,f(x) = \tan(k_2x),\; k_2\ne0 \,$ and thus, does not qualify.

I am interested in those which are
satisfied by **both** $f=\sin$ and $f=\tan$.

In all but **one** of the identities of this kind that I know of,
they are **also** satisfied by $\,f(x)=k_1\text{sn}(k_2\,x|m),\,$
the Jacobi elliptic function $\,\text{sn},\,$ as well as the related
$\,\text{sc}, \text{sd}\,$ functions. The one exception is for an
identity with Jacobi Zeta and Epsilon function solutions.
This leads to two natural questions.

1.Do identities exist with solutions aside from the Jacobi functions mentioned?

2.Do identities exist withonlysine and tangent solutions?

NOTE: Perhaps it would be easier to understand a specialization case. Suppose there is only one variable $\,a.\,$ Consider the polynomial ring $\,\mathbb{Z}[f_1,f_2,f_3,\dots].\,$ In the first functional equation $(1)$ replace $\,b\,$ with $\,2a,\,$ and $\,c\,$ with $\,-2a\,$ to get the equation

$$ f(a)f(3a)f(4a)-f(2a)^2f(4a)+f(a)f(2a)f(3a)-f(a)^2f(2a) = 0.\tag{5} $$
The polynomial equation associated with this equation is
$$ f_1f_3f_4-f_2^2f_4+f_1f_2f_3-f_1^2f_2 =0 \tag{6}$$
where $\,f_n:=f(na).$
This single polynomial equation **also** has solutions
$\,f(x)=k_1\text{sn}(k_2\,x|m)\,$ and
seems to be the simplest such equation for the Jacobi sn function.
There are an infinite number of other equations which come from
specializing the first functional equation $(1)$. I conjecture
that there is some kind of basis for the ideal of all such
equations. The issues raised here are similar to the ones for my
"Dedekind Eta-function Identities" list and studied by Ralf
Hemmecke in his 2018 article
"Construction of all polynomial relations among Dedekind eta functions of level N".

NOTE: The 18 identities I refer to are in my file Special Algebraic Identities (ident04.txt) along with hundreds of special algebraic identities (also available via the Wayback Machine).