题目

α＋β=2π/3,求证2cos^2(α)-sin^2(β)=cos(2α+π/3)

提问时间：2021-03-25

答案

左边=2[cos^2α-sin^2β]
=2[(1+cos2α)/2-(1-cos2β)/2]
=cos2a+cos2β
=cos2α+(cos(4π/3-2α)
=cos2α+[cos(4π/3)cos2α+sin(4π/3)sin2α]
=cos2α+[-cos2a/2-√3sin2α/2]
=cos2α-cos2α/2-√3sin2α/2
=cos2α/2-√3sin2α/2
右边=cos(2α+π/3)
=cos2αcosπ/3-sin2αsinπ/3.
=cos2α/2-√3sin2α/2
左边等于右边,得证.
说明一下,你要证明的等式应该是：2[cos^2(α)-sin^2(β)]=cos(2α+π/3),掉了一个中括号,对吧?