Question

xnnn

想問第二題是怎麼從 1+cos^2 75-sin^2 15 變成 1+cos(75+15)cos(75-15)的？

(1) sin² 37.5°-sin27.50。图
sin2 75° -sin2 150
(2)
。
仄
cos2 75°+cos2 150
= 1-454-15
-7I+ 604-75 -4th " 15
2
* 414 9 6² = 1
6390"20
STU 13715+7.5) 4th (37.5-7.5)
5Th 45 STU 30 = Ex = = 12/
(2)
5Th 195415) ST17515)
1+ 403(75+15)405175-15)
=
=
D) net (1)
)(S)
B+B (E)
1+0x吾
匡
=

Answer 1

qn

sin和cos的平方差公式
sin²α - sin²β = cos²β - cos²α = sin(α+β)sin(α-β)
cos²α - sin²β = cos²β - cos²α = cos(α+β)cos(α-β)

(從後面往前推導比較順，否則要憑空變出東西不好想)
cos²α - sin²β
= cos²α - cos²α sin²β - sin²β + cos²α sin²β
= cos²α (1-sin²β) - sin²β (1-cos²α)
= cos²α cos²β - sin²α sin²β
= (cosα cosβ - sinα sinβ)(cosα cosβ + sinα sinβ)
= cos(α+β)cos(α-β)