sin78°-sin42°=sin(60°+18°)-sin(60°-18°)= 2cos60°sin18°=sin18°
sin66°-sin6°=sin(60°+6°)-sin6°=sin60°cos6°+cos60°sin6°-sin6°
=sin(60°-6°)=sin54°
∴sin78°-sin66°-sin42°+sin6°=sin78°-sin42°-(sin66°-sin6°)
=sin18°-sin54°=sin18°-cos(2*18°)=sin18°-1+2sin²18°
下面求sin18°
令x = 18°
∴cos3x = sin2x
∴4(cosx)^3 - 3cosx = 2sinxcosx
∵cosx≠ 0
∴4(cosx)^2 - 3 = 2sinx
∴4sinx² + 2sinx - 1 = 0,
∴sin18°+2sin²18°=1/2
∴原式=1/2-1=-1/2