Prove that: `sin^2pi/(18)+sin^2pi/9+sin^2(7pi)/(18)+sin^2(4pi)/9=2`

Explanation

`sin^2pi/(18)+sin^2pi/9+sin^2(7pi)/(18)+sin^2(4pi)/9`

`=sin^2pi/(18)+sin^2(2pi)/9+sin^2(7pi)/(18)+sin^2(8pi)/18`

`=sin^2 (pi/18)+sin^2 frac{2pi}{18}+sin^2 frac{7pi}{18}+sin^2(frac{8pi}{18})`

`=sin^2 (pi/18)+sin^2 frac{2pi}{18}+sin^2(frac{pi}{2}-frac{2pi}{18})+sin^2(frac{pi}{2}-frac{pi}{18})`

`=sin^2 frac{pi}{18} cos^2frac{pi}{18}+sin^2 frac{2pi}{18}+cos^2frac{2pi}{18}`

`=1+1`

`=2`