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 ?


$$\frac{\sin 80^\circ \sin 20^\circ\sin 100^\circ }{\sin 50^\circ}=\frac{\sin 80^\circ \sin 20^\circ\cdot 2\sin 50^\circ \cos 50^\circ }{\sin 50^\circ}=2\sin 80^\circ\sin 20^\circ\cos 50^\circ=2\sin 80^\circ\sin 20^\circ\sin40^\circ$$
as $\cos 50^\circ=\sin40^\circ$
Now by Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $, $$\sin\left(60^\circ-x\right)\sin\left(60^\circ+x\right)=\sin^260^\circ-\sin^2x$$
$$\implies\sin x\sin\left(60^\circ-x\right)\sin\left(60^\circ+x\right)=\dfrac{\sin3x}4$$
Here $x=20^\circ$
