Question

证明sin(a+b)sin(a-b)=sin²a-sin² b，并用该式计算sin²20°+sin80°sin40°

Answer 1

sin(a+b)sin(a-b)
=(sinacosb+sinbcosa)(sinacosb-sinbcosa)
=(sinacosb)^2+sinasinbcosacosb-sinasinbcosacosb-(sinbcosa)^2
=(sinacosb)^2-(1-sin^2a)(1-cos^2b)
=(sinacosb)^2-[1-sin^2a-cos^2b+(sinacosb)^2]
=-1+sin^2a+(1-sin^2b)
=sin^2 a-sin^2 b

sin²20°+sin80°sin40°
=sin²20°+sin(60°+20°)sin(60°-20°)
=sin²20°+sin²60°-sin²20°
=sin²60°
=(√3/2)²
=3/4