Originally Posted by
red_dog The equality is equivalent with
$\displaystyle \sin 20+4\sin 20\cos 20=\sqrt{3}\cos 20\Leftrightarrow$
$\displaystyle \Leftrightarrow \sin 20+2\sin 40=\sqrt{3}\cos 20\Leftrightarrow$
$\displaystyle \Leftrightarrow \sin 40=\frac{\sqrt{3}}{2}\cos 20-\frac{1}{2}\sin 20\Leftrightarrow$
$\displaystyle \Leftrightarrow \sin 40=\cos 30\cos 20-\sin 30\sin 20\Leftrightarrow$
$\displaystyle \Leftrightarrow \sin 40=\cos 50\Leftrightarrow$
$\displaystyle \Leftrightarrow \sin 40=\sin 40$