cos²15°－sin²15°＝?

题目

cos²15°－sin²15°＝?

答案

方法一:
(cos^2)15°-(sin^2)15
=(cos15+sin15)(cos15-sin15)
=[sin(90-15)+sin15][sin(90-15)-sin15]
=(sin75+sin15)(sin75-sin15)
=2*sin(75+15)/2*cos(75-15)/2*2*cos(75+15)*sin(75-15)/2
=4*sin45*cos30*cos45*sin30
=√3/2.
方法二:
cos15°=cos(45-30)=cos45*cos30+sin45*sin30
=√2/2*√3/2+√2/2*1/2
=(√6+√2)/4.
cos^2(15)=[(√6+√2)/4]^2=(2+√3)/4,
sin15=sin(45-30)=sin45*cos30-cos45*sin30
=√2/2*√3/2-√2/2*1/2
=(√6-√2)/4.
sin^2(15)=[(√6-√2)/4]^2=(2-√3)/4.
(cos^2)15°-(sin^2)15=(2+√3)/4-(2-√3)/4
=√3/2.