$\mathrm{Use\:the\:following\:identity}:\quad \cos \left(s\right)\cos \left(t\right)+\sin \left(s\right)\sin \left(t\right)=\cos \left(s-t\right)\\\sin \left(\frac{2\pi }{9}\right)\sin \left(\frac{\pi }{18}\right)+\cos \left(\frac{2\pi }{9}\right)\cos \left(\frac{\pi }{18}\right)=\cos \left(\frac{2\pi }{9}-\frac{\pi }{18}\right)\\=\cos \left(\frac{2\pi }{9}-\frac{\pi }{18}\right)\\\mathrm{Join}\:\frac{2\pi }{9}-\frac{\pi }{18}:\quad \frac{\pi }{6}\\\frac{2\pi }{9}-\frac{\pi }{18}\\$

$\mathrm{Least\:Common\:Multiplier\:of\:}9,\:18:\quad 18\\Adjust\:Fractions\:based\:on\:the\:LCM\\=\frac{4\pi }{18}-\frac{\pi }{18}\\\mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}:\quad \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}\\=\frac{4\pi -\pi }{18}\\4\pi -\pi =3\pi \\4\pi -\pi =3\pi \\\mathrm{Cancel\:the\:common\:factor:}\:3\\=\frac{\pi }{6}\\=\cos \left(\frac{\pi }{6}\right)\\$

$Answer: \mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$

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3 years ago