Found this equality $$\tan 15^{\circ} = \tan 27^{\circ} \tan 33^{\circ} \tan 39^{\circ}$$ with a random search. It checks with WA, and these kind of equalities can be proved automatically.
I am looking for an elementary proof.
Some similarly looking equalities are using the identity;
$$\tan 3 x= \tan x\cdot \tan(\frac{\pi}{3}-x)\cdot \tan (\frac{\pi}{3}+x)$$
We notice that $27^{\circ} + 33^{\circ} = 60^{\circ}$, but does not seem enough.
Thank you for interest!
$\bf{Added:}$ Let me add the mechanical proof that works like here. One considers the basic angle $x = 3^{\circ}= \frac{2 \pi}{120}$, and we need to show that $\tan 5 x= \tan 9 x \tan 11 x \tan 13 x$, which is equivalent to: $z= e^{\frac{2 \pi i}{120}}$ is a root of the equation
$$\frac{z^{10}-1}{z^{10}+1} + \frac{z^{18} -1}{z^{18} + 1}\cdot \frac{z^{22} -1}{z^{22} + 1}\cdot \frac{z^{26} -1}{z^{26} + 1}=0$$
LHS factors and we see the cyclotomic polynomial $\Phi_{120}(z)$ in the numerator.
$\bf{Added:}$ Since in general we have to show some equality for functions of multiples of an angle $x$, where $x$ is a rational multiple of $2\pi$, we could prove an equivalent equality for the angle $m x$, where $m$ is relatively prime to the denominator of $\frac{x}{2 \pi}$, so we might reduce to other known identities. Perhaps this one works with say $m = 7$? Since $7\times 15^{\circ} = 90^{\circ} + 15^{\circ}$.