tan189°-cot63°+tan297°-cot171°
=tan(180°+9°)-cot63°+tan(360°-63°)-cot(180°-9°)
=tan9°-cot63°-tan63°+cot9°
=sin9°/cos9°+cos9°/sin9°-cos63°/sin63°-sin63°/cos63°[将上步的第四项放到第二项]
=[(sin9°)^2+(cos9°)^2]/(sin9°cos9°)-[(cos63°)^2+(sin63°)^2]/(sin63°cos63°) [前两项、后两项分别通分]
=2/sin18°-2/sin126°[分母分别用了2倍角公式sinαcosα=(1/2)sin2α]
=2/sin18°-2/sin(180°-54°)
=2/sin18°-2/sin54°
=2(sin54°-sin18°)/(sin18°sin54°)
=2[sin(36°+18°)-sin(36°-18°)]/[sin18°cos(90°-54°)]
=4(cos36°sin18°)/(sin18°cos36°)[分子中用和差角公式展开并且化简]
=4
注意：
(1)二楼的解法用到了“和差化积”公式,这超出了高考要求.
(2)直接用公式sin(α+β)+sin(α-β)=2sinαcosβ,和sin(α+β)-sin(α-β)=2cosαsinβ可以简化运算,又避免了“和差化积”公式.