Докажите равенство sin200+sin100=sin40

Решите задачу:

$sin200+sin100=sin40\\\\ 2sin100cos100+sin100=sin100(2cosa100+1)\\\\ sin(60+40)(2cos(60+40)+1)\\\\ (\frac{\sqrt{3}*cos40+sin40}{2})(cos40-\sqrt{3}sin40+1)\\\\ \frac{-2cos40sin40+\sqrt{3}*cos40+\sqrt{3}*cos^240+sin40-\sqrt{3}*sin^240}{2}\\\\ \frac{-2cos40sin40+\sqrt{3}cos40+\sqrt{3}*cos80+sin40}{2}\\\\ \frac{-2cos40sin40+\sqrt{3}(cos40+cos80)+sin40}{2}\\\\ \frac{-2cos40*2sin20*cos20+\sqrt{3}cos(20)}{2}\\\\ \frac{cos20(-4cos40*sin20+\sqrt{3})+sin40}{2}\\\\ \frac{cos20(-2(-sin20+\frac{\sqrt{3}}{2}+\sqrt{3})}{2}$
$\frac{sin40+sin40}{2}=sin40$