3csc＾2 20°-sec＾2 20°=32cos40°

证明：
3csc＾2 20°-sec＾2 20°
＝3/(sin20°)^2-1/(cos20°)^2
=[3(cos20°)^2-(sin20°)^2]/(sin20°cos20°)^2（下一步分子降次、分母用2倍角正弦公式）
=[(3/2)(1+cos40°)-(1/2)(1-cos40°)]/(1/4)(sin40°)^2
=8(0.5+cos40°)/(sin40°)^2
=8(cos60°+cos40°)/(sin40°)^2
=8[cos(50°+10°)+cos(50°-10°)]/(sin40°)^2
=16con50°cos10°/(sin40°)^2
=16cos10°/sin40°
=16sin80°/sin40°
=32sin40°cos40°/sin40°
=32cos40°.