How can the trigonometric equation be proven? This question :
https://math.stackexchange.com/questions/1411700/whats-the-size-of-the-x-angle
has the answer $10°$. This follows from the equation
$$2\sin(80°)=\frac{\sin(60°)}{\sin(100°)}\times \frac{\sin(50°)}{\sin(20°)}$$
which is indeed true , which I checked with Wolfram.


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*How can this equation be proven ?


$$\begin{align}\frac{\sin 80^\circ \sin 20^\circ\sin 100^\circ }{\sin 50^\circ}&=\frac{\sin 80^\circ \sin 20^\circ\cdot 2\sin 50^\circ \cos 50^\circ }{\sin 50^\circ}\\&=2\sin 80^\circ\sin 20^\circ \cos 50^\circ\\&=2\left(-\frac 12(\cos 100^\circ-\cos 60^\circ)\right)\cos 50^\circ\\&=(\cos 60^\circ-\cos 100^\circ)\cos 50^\circ\\&=\frac 12\cos 50^\circ-\cos 100^\circ\cos 50^\circ\\&=\frac 12\cos 50^\circ-\frac 12(\cos 150^\circ+\cos 50^\circ)\\&=-\frac 12\times\left(-\frac{\sqrt 3}{2}\right)\\&=\frac{\sqrt 3}{4}\\&=\frac 12\sin 60^\circ\end{align}$$
