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求证：sin3a*cos^3a+cos3a*sin^3a=3/4sin4a
题目详情
求证：sin3a*cos^3a+cos3a*sin^3a=3/4sin4a
▼优质解答
答案和解析
sin3a
=sin(2a+a)
=sin2acosa+cos2asina
=2sina(1-sin^2a)+(1-2sin^2a)sina
=3sina-4sin^3a
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos^2a-1)cosa-2(1-cos^a)cosa
=4cos^3a-3cosa
sin3a*cos^3a+cos3a*sin^3a
=(3sina-4sin^3a )*cos^3a+(4cos^3a-3cosa )*sin^3a
=3sina*cos^3a-4sin^3a *cos^3a+4cos^3a*sin^3a-3cosa*sin^3a
=3sina*cos^3a-3cosa*sin^3a
=3sina*cosa(cos^2a-sin^2a)
=3/2*sin2a*cos2a
=3/2*1/2*sin4a
=3/4 sin4a
=sin(2a+a)
=sin2acosa+cos2asina
=2sina(1-sin^2a)+(1-2sin^2a)sina
=3sina-4sin^3a
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos^2a-1)cosa-2(1-cos^a)cosa
=4cos^3a-3cosa
sin3a*cos^3a+cos3a*sin^3a
=(3sina-4sin^3a )*cos^3a+(4cos^3a-3cosa )*sin^3a
=3sina*cos^3a-4sin^3a *cos^3a+4cos^3a*sin^3a-3cosa*sin^3a
=3sina*cos^3a-3cosa*sin^3a
=3sina*cosa(cos^2a-sin^2a)
=3/2*sin2a*cos2a
=3/2*1/2*sin4a
=3/4 sin4a
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