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

數(shù)學(xué)人氣：130 ℃時(shí)間：2020-09-18 20:02:59

優(yōu)質(zhì)解答

左邊=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
右邊=cos(2α+π/3)
=cos2αcosπ/3-sin2αsinπ/3.
=cos2α/2-√3sin2α/2
左邊等于右邊,得證.
說明一下,你要證明的等式應(yīng)該是：2[cos^2(α)-sin^2(β)]=cos(2α+π/3),掉了一個(gè)中括號(hào),對吧?