Question

（1）求证：sin（A+B）sin（A-B）=sin^2A-sin^2B

（2）利用（1）的结论求值：sin^220°+sin10°sin50°

Answer 1

用sinAsinB=1/2[cos(A-B)-cos(A+B)]
所以sin（A+B）sin（A-B）=1/2(cos2A-cos2B)
cos2A=1-2sin^A,cos2B=1-2sin^B
代入得证．

Answer 2

sin（A+B）sin（A-B）=(sinAcosB+cosAsinB)(sinAcosB-cosAsinB)
=sin^2Acos^2B-cos^2Asin^2B
=sin^2Acos^2B-(1-sin^2A)sin^2B
=sin^2Acos^2B-sin^2B+sin^2Bsin^2A
=sin^2A-sin^2B
sin10°sin50° =sin^30-sin^220
所以原式=sin^230=1/4

Answer 3

送佛送到西（不是夸你是佛哦）
sin^220°+sin10°sin50＝sin^40+sin(30+20)sin(30-20)=sin^40+sin^60-sin^40=3/4

Answer 4

不好意思，这步应为sin（A+B）sin（A-B）=1/2(cos2Ｂ-cos2Ａ)