Why is $\tan 54^\circ\approx \frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$

Question

This question was asked as an equality on MSE and I am quite surprised to find that its strictly false

However I would like to see why is their difference of the order $10^{-15}$?

$$\tan 54^\circ\approx \dfrac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$$

Answer 1

$\tan 54^\circ\color{red}{=} \frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$ is true as darth geek proved analytically. The difference of $10^{-15}$ is a numerical error.

EDIT: One can prove this directly by using the exact values of sine, cosine and tangent.