sin²10°+sin²20°+sin²30°+……+sin²80°=?

sin²10°+sin²20°+sin²30°+……+sin²80°
=sin²10°+sin²20°+sin²30°+sin²40°+sin²50°+sin²60°+sin²70+sin²80°
=sin²10°+sin²20°+sin²30°+sin²40°+cos²40°+cos²30°+cos²20+cos²10°
=sin²10°+cos²10°+sin²20°+cos²20+sin²30°+cos²30°+sin²40°+cos²40°
=1+1+1+1
=4

(sin²10°+sin²20°+sin²30°+……+sin²80°)+(sin²10°+sin²20°+sin²30°+……+sin²80°)
=(sin²10°+sin²20°+sin²30°+……+sin²80°)+(cos²80°+cos²70°+cos²60°+……+cos²10°)
=(sin²10°+cos²10°)+(sin²20°+cos²20°)+……+(sin²80°+cos²80°)
=1*8
=8
∴sin²10°+sin²20°+sin²30°+……+sin²80°=8/2=4

sin²10°+sin²20°+sin²30°+……+sin²80°
=cos²80°+cos²70°+cos²60°+cos²50°+sin²50°+sin²60°+sin²70°+sin²80°
=cos²80°+sin²80°+cos²70°+sin²70°+cos²60°+sin²60°+cos²50°+sin²50°
=1+1+1+1
=4

sin80=cos10